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Writing Quadratic Equations from Real Solutions

7.13.4 Writing Quadratic Equations from Real Solutions

7.13.4 • Writing Quadratic Equations from Real Solutions

For questions 1 – 4, use the following information:

A parabola crosses the x x -axis at (-1,0) and (2,0). We want to write a quadratic equation with those x x -intercepts. We can do this without graphing.
1.

When y = 0 y = 0 , what must x x be?

Compare your answer:

We know that (-1,0) and (2,0) are points on the graph when y = 0 y = 0 . So there are two possibilities for x x : x = − 1 x = − 1 and x = 2 x = 2 .

x = − 1 x = − 1 and x = 2 x = 2 are called the real solutions of the quadratic because they are the solutions of the equation when we set y = 0 y = 0 .

2.

Rewrite the equations from number 1 so they both equal zero.

Compare your answer:

x + 1 = 0 x + 1 = 0 and x − 2 = 0 x − 2 = 0 .

Because each of the equations equals zero, multiplying them together will also equal zero.

3.

What equation do you get when you multiply the previous equations together?

Compare your answer:

( x + 1 ) ( x − 2 ) = 0 ( x + 1 ) ( x − 2 ) = 0

4.

Multiply the binomials.

Compare your answer:

x 2 − x − 2 = 0 x 2 − x − 2 = 0

This answer can also give us the more general quadratic with those x x -intercepts. We replace 0 with y y to show that y y varies and isn’t always 0. So, y = x 2 − x − 2 y = x 2 − x − 2 is a parabola that crosses the x x -axis at -1 and 2.

Self Check

The solution to a quadratic equation is x = 4 x = 4 and x = − 8 x = − 8 . What is the equation?

x2 + 32x − 12=0 x2 + 32x − 12=0
x2 + 4x − 32=0 x2 + 4x − 32=0
x2 − 32x + 4=0 x2 − 32x + 4=0
x2 − 4x + 32=0 x2 − 4x + 32=0

Additional Resources

When you write a quadratic in factored form, that helps you find the x x -intercepts.

y = x 2 + 2 x − 3 y = x 2 + 2 x − 3

y = ( x + 3 ) ( x − 1 ) y = ( x + 3 ) ( x − 1 )

When we set y = 0 y = 0 , we find the places where the parabola crosses the x x -axis.

0 = ( x + 3 ) ( x − 1 ) 0 = ( x + 3 ) ( x − 1 )

So, 0 = x + 3 0 = x + 3 and 0 = x − 1 0 = x − 1

x = − 3 x = − 3 and 1.

-3 and 1 are called the real solutions to the quadratic equation.

We can also start with the real solutions and use them to write quadratic equations.

When we know that x = − 3 x = − 3 and 1, then that means 0 = x + 3 0 = x + 3 and 0 = x − 1 0 = x − 1 .

So, 0 = ( x + 3 ) ( x − 1 ) 0 = ( x + 3 ) ( x − 1 ) .

And, using the FOIL method to multiply this out, we come to

0 = x 2 + 2 x − 3 . 0 = x 2 + 2 x − 3 .

When y y is not equal to zero, we have the following quadratic equation

y = x 2 + 2 x − 3 y = x 2 + 2 x − 3 , which is exactly where we started!

Try it

Writing Quadratic Equations from Real Solutions

1.

Write a quadratic from the real solutions x = 1 x = 1 and x = 7 x = 7 without graphing.

Compare your answer: x 2 − 8 x + 7 x 2 − 8 x + 7

x = 1 x = 1 and x = 7 x = 7 become x − 1 = 0 x − 1 = 0 and x − 7 = 0 x − 7 = 0 . y = ( x − 1 ) ( x − 7 ) = x 2 − x − 7 x + 7 = x 2 − 8 x + 7 y = ( x − 1 ) ( x − 7 ) = x 2 − x − 7 x + 7 = x 2 − 8 x + 7 .