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Solving quadratic equations | Lesson

A guide to solving quadratic equations on the digital SAT

What are quadratic equations?

A quadratic equation is an equation with a
variable to the second power
as its highest power term. For example, in the quadratic equation 3, x, squared, minus, 5, x, minus, 2, equals, 0:
  • x is the variable, which represents a number whose value we don't know yet.
  • The squared is the power or exponent. An exponent of 2 means the variable is
    multiplied by itself
    .
  • 3 and minus, 5 are the coefficients, or constant multiples of x, squared and x. 3, x, squared is a single
    term
    , as is minus, 5, x.
  • minus, 2 is a constant term.
In this lesson, we'll learn to:
  1. Solve quadratic equations in several different ways
  2. Determine the number of solutions to a quadratic equation without solving
You can learn anything. Let's do this!

How do I solve quadratic equations using square roots?

Solving quadratics by taking square roots

Khan Academy video wrapper
Solving quadratics by taking square rootsSee video transcript

When can I solve by taking square roots?

Quadratic equations without x-terms such as 2, x, squared, equals, 32 can be solved without setting a quadratic expression equal to 0. Instead, we can isolate x, squared and use the square root operation to solve for x.
When solving quadratic equations by taking square roots, both the positive and negative square roots are solutions to the equation. This is because when we square a solution, the result is always positive.
For example, for the equation x, squared, equals, 4, both 2 and minus, 2 are solutions:
  • 2, squared, equals, start superscript, \checkmark, end superscript, 4
  • left parenthesis, minus, 2, right parenthesis, squared, equals, start superscript, \checkmark, end superscript, 4
When solving quadratic equations without x-terms:
  1. Isolate x, squared.
  2. Take the square root of both sides of the equation. Both the positive and negative square roots are solutions.

Example: What values of x satisfy the equation 2, x, squared, equals, 18 ?

Try it!

TRY: identify the steps to solving a quadratic equation
x, squared, minus, 3, equals, 13
We can solve the quadratic equation above by first
both sides of the equation, which gives us the equation x, squared, equals, 16.
Next, we can take the square root of both sides of the equation, which gives us the solution(s)
.


What is the zero product property, and how do I use it to solve quadratic equations?

Zero product property

Khan Academy video wrapper
Zero product propertySee video transcript

Zero product property and factored quadratic equations

The zero product property states that if a, b, equals, 0, then either a or b is equal to 0.
The zero product property lets us solve factored quadratic equations by solving two linear equations. For a quadratic equation such as left parenthesis, x, minus, 5, right parenthesis, left parenthesis, x, plus, 2, right parenthesis, equals, 0, we know that either x, minus, 5, equals, 0 or x, plus, 2, equals, 0. Solving these two linear equations gives us the two solutions to the quadratic equation.
To solve a factored quadratic equation using the zero product property:
  1. Set each factor equal to 0.
  2. Solve the equations from Step 1. The solutions to the linear equations are also solutions to the quadratic equation.

Example: What are the solutions to the equation left parenthesis, x, minus, 4, right parenthesis, left parenthesis, 3, x, plus, 1, right parenthesis, equals, 0 ?

Try it!

TRY: use factors to determine the solutions
2, x, squared, plus, x, minus, 3, equals, left parenthesis, x, minus, 1, right parenthesis, left parenthesis, 2, x, plus, 3, right parenthesis
The equation above shows the factors of the quadratic expression 2, x, squared, plus, x, minus, 3. Which of the following equations, when solved, give us the solutions to the equation 2, x, squared, plus, x, minus, 3, equals, 0 ?
Choose all answers that apply:
Choose all answers that apply:


How do I solve quadratic equations by factoring?

Solving quadratics by factoring

Khan Academy video wrapper
Solving quadratics by factoringSee video transcript

Solving factorable quadratic equations

If we can write a quadratic expression as the product of two linear expressions (factors), then we can use those linear expressions to calculate the solutions to the quadratic equation.
In this lesson, we'll focus on factorable quadratic equations with 1 as the coefficient of the x, squared term, such as x, squared, minus, 2, x, minus, 3, equals, 0. For more advanced factoring techniques, including special factoring and factoring quadratic expressions with x, squared coefficients other than 1, check out the Factoring quadratic and polynomial expressions lesson.
Recognizing factors of quadratic expressions takes practice. The factors will be in the form left parenthesis, x, plus, a, right parenthesis, left parenthesis, x, plus, b, right parenthesis, where a and b fulfill the following criteria:
  • The sum of a and b is equal to the coefficient of the x-term in the unfactored quadratic expression.
  • The product of a and b is equal to the constant term of the unfactored quadratic expression.
For example, we can solve the equation x, squared, minus, 2, x, minus, 3, equals, 0 by factoring x, squared, minus, 2, x, minus, 3 into left parenthesis, x, plus, a, right parenthesis, left parenthesis, x, plus, b, right parenthesis, where:
  • a, plus, b is equal to the coefficient of the x-term, minus, 2.
  • a, b is equal to the constant term, minus, 3.
minus, 3 and 1 would work:
  • minus, 3, plus, 1, equals, minus, 2
  • left parenthesis, minus, 3, right parenthesis, left parenthesis, 1, right parenthesis, equals, minus, 3
This means we can rewrite x, squared, minus, 2, x, minus, 3, equals, 0 as left parenthesis, x, minus, 3, right parenthesis, left parenthesis, x, plus, 1, right parenthesis, equals, 0 and solve the quadratic equation using the zero product property. Keep mind that a and b are not themselves solutions to the quadratic equation!
When solving factorable quadratic equations in the form x, squared, plus, b, x, plus, c, equals, 0:
  1. Rewrite the quadratic expression as the product of two factors. The two factors are linear expressions with an x-term and a constant term. The sum of the constant terms is equal to b, and the product of the constant terms is equal to c.
  2. Set each factor equal to 0.
  3. Solve the equations from Step 2. The solutions to the linear equations are also solutions to the quadratic equation.

Example: What are the solutions to the equation x, squared, plus, 4, x, minus, 5, equals, 0 ?

Try it!

Try: match the equivalent quadratic expressions
Match each factored expression to its equivalent unfactored expression in the table below.


How do I use the quadratic formula?

The quadratic formula

Khan Academy video wrapper
The quadratic formulaSee video transcript

Using the quadratic formula to solve equations and determine the number of solutions

Not all quadratic expressions are factorable, and not all factorable quadratic expressions are easy to factor. The quadratic formula gives us a way to solve any quadratic equation as long as we can plug the correct values into the formula and evaluate.
For start color #7854ab, a, end color #7854ab, x, squared, plus, start color #ca337c, b, end color #ca337c, x, plus, start color #208170, c, end color #208170, equals, 0:
x, equals, start fraction, minus, start color #ca337c, b, end color #ca337c, plus minus, square root of, start color #ca337c, b, end color #ca337c, squared, minus, 4, start color #7854ab, a, end color #7854ab, start color #208170, c, end color #208170, end square root, divided by, 2, start color #7854ab, a, end color #7854ab, end fraction
Note: the quadratic formula is not provided in the reference section of the SAT! You'll have to memorize the formula to use it.

What are the steps?

To solve a quadratic equation using the quadratic formula:
  1. Rewrite the equation in the form a, x, squared, plus, b, x, plus, c, equals, 0.
  2. Substitute the values of a, b, and c into the quadratic formula, shown below.
x, equals, start fraction, minus, b, plus minus, square root of, b, squared, minus, 4, a, c, end square root, divided by, 2, a, end fraction
  1. Evaluate x.

Example: What are the solutions to the equation x, squared, minus, 6, x, equals, 9 ?

The b, squared, minus, 4, a, c portion of the quadratic formula is called the discriminant. The value of b, minus, 4, a, c tells us the number of unique real solutions the equation has:
  • If b, squared, minus, 4, a, c, is greater than, 0, then square root of, b, squared, minus, 4, a, c, end square root is a real number, and the quadratic equation has two real solutions, start fraction, minus, b, minus, square root of, b, squared, minus, 4, a, c, end square root, divided by, 2, a, end fraction and start fraction, minus, b, plus, square root of, b, squared, minus, 4, a, c, end square root, divided by, 2, a, end fraction.
  • If b, squared, minus, 4, a, c, equals, 0, then square root of, b, squared, minus, 4, a, c, end square root is also 0, and the quadratic formula simplifies to start fraction, minus, b, divided by, 2, a, end fraction, which means the quadratic equation has one real solution.
  • If b, squared, minus, 4, a, c, is less than, 0, then square root of, b, squared, minus, 4, a, c, end square root is an imaginary number, which means the quadratic equation has no real solutions.

Try it!

Try: set up for the quadratic formula
7, x, squared, plus, 6, x, minus, 1, equals, 0
If we want to use the quadratic formula to solve the equation above, what are the values of a, b, and c we should plug into the quadratic formula?
a, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
b, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
c, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text


Try: use the discriminant to find the number of solutions
7, x, squared, plus, 6, x, minus, 1, equals, 0
The value of the
discriminant
for the quadratic equation above is
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
.
Because the discriminant is
, the equation has
.


Try: substitute into the quadratic formula
7, x, squared, plus, 6, x, minus, 1, equals, 0
Which of the following expressions, when evaluated, gives the solutions to the equation above?
Choose 1 answer:
Choose 1 answer:


Your turn!

Practice: Solve quadratic equation using square root
If start fraction, 1, divided by, 2, end fraction, x, squared, equals, 32 and x, is greater than, 0, what is the value of x ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text


Practice: Solve quadratic equation by factoring
x, squared, plus, x, minus, 56, equals, 0
What are the solutions to the equation above?
Choose 1 answer:
Choose 1 answer:


Practice: Solve quadratic equation with the quadratic formula
Which of the following values of x satisfy the equation minus, 3, x, squared, plus, 12, x, plus, 4, equals, 0 ?
Choose 1 answer:
Choose 1 answer:


Practice: Determine the condition for one real solution
If a, x, squared, plus, 8, x, plus, 1, equals, 0, for what value of a does the equation have exactly one real solution?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text


Things to remember

For a, x, squared, plus, b, x, plus, c, equals, 0:
x, equals, start fraction, minus, b, plus minus, square root of, b, squared, minus, 4, a, c, end square root, divided by, 2, a, end fraction
  • If b, squared, minus, 4, a, c, is greater than, 0, then the equation has 2 unique real solutions.
  • If b, squared, minus, 4, a, c, equals, 0, then the equation has 1 unique real solution.
  • If b, squared, minus, 4, a, c, is less than, 0, then the equation has no real solution.

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