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 Depdendent Variable

 Number of equations to solve: 23456789
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 Dependent Variable

 Number of inequalities to solve: 23456789
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Example

The solutions of a quadratic equation are 5 and

Work backwards to find a quadratic equation with these solutions.

Solution

 Begin by writing the two solutions. To clear the fraction in the second solution, multiply both sides of the second equation by 3. x = 5 or 3x = -2 Move the constant to the left side of each equation. x - 5 = 0 or 3x + 2 = 0 Since each binomial is equal to 0, their product is 0. (x - 5)(3x + 2) = 0 Multiply the binomials. Combine like terms. 3x2 + 2x - 15x - 10 = 0 3x2 - 13x - 10 = 0

The quadratic equation 3x2 - 13x - 10 = 0 has the given solutions, 5 and

Note:

3x2 - 13x - 10 = 0

Any nonzero multiple of this equation also has solutions 5 and

Here are some examples:

6x2 - 26x - 20 = 0

15x2 - 65x - 50 = 0

The quadratic formula states that the solutions of a quadratic equation, ax2 + bx + c = 0, are

Letâ€™s see what happens when we combine the solutions first by addition and then by multiplication:

 â€¢ When we add the solutions, the radicals are eliminated. Add the numerators to form one fraction. Combine like terms. The square root terms add to zero. Cancel the common factor, 2.
Thus, for a quadratic equation, ax2 + bx + c = 0, the sum of the solutions is

â€¢ When we multiply the solutions, the radicals are eliminated.

Multiply the numerators and multiply the denominators.

Combine like terms.

Cancel the common factor, 4a.

Thus, for a quadratic equation, ax2 + bx + c = 0, the product of the solutions is