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Quadratic Equations

Example 1

The solutions of a quadratic equation are 4 and -1. Use their sum and product to find a quadratic equation with those solutions.

Solution

For a quadratic equation, ax² + bx + c = 0, the sum of the solutions is

The sum of the given solutions is

We have

So, if a = 1, then b = -3.

The product of the solutions is

The product of the given solutions is

We have

So, if a = 1, then c = -4.

For the quadratic equation, ax² + bx + c = 0, let a = 1, b = -3, and c = -4.	ax² + bx + c = 0
Substitute these values in the equation. Simplify.	1x² + (-3)x - 4 = 0 x² - 3x - 4 = 0

The equation, x² - 3x - 4 = 0 is a quadratic equation with solutions 4 and -1.

Note:

x² - 3x - 4 = 0

Any nonzero multiple of this equation also has solutions 4 and -1.

Here are some examples:

2x² - 6x - 8 = 0

3x² - 9x - 12 = 0

Example 2

The quadratic equation 2x² - 7x + c = 0 has discriminant 9.

What are the solutions of the equation?

Solution

First, weâ€™ll find the value of c. Then weâ€™ll solve the equation.

To find c, we use the discriminant, b² - 4ac. For the given equation, a = 2 and b = -7. We also know the discriminant is 9. Substitute a = 2 and b = -7. Simplify. Subtract 49 from both sides. Divide both sides by -8. Now we know that c = 5.	b² - 4ac = 9 (-7)² - 4(2)c = 9 49 - 8c = 9 -8c = -40 c = 5
We use the quadratic formula to solve the equation 2x² - 7x + c = 0.
Substitute a = 2, b = -7, and c = 5.
Simplify.
Simplify the radicand.
Simplify the square root.
Simplify.

Thus, if the discriminant of 2x² - 7x + c = 0 is 9, the solutions of the equation are

and 1.