The quadratic formula is used to find, roots, or zeroes, to quadratic functions when the equation isn't factorable and solving for x when y = 0 is too difficult. This formula also gives the x-value of the vertex and the discriminant provides the number of solutions.

For any quadratic equation of the form y = ax2 + bx + c, the quadratic formula below

$x=\frac{-b±\sqrt{{b}^{2}-4ac}}{2a}$

will find the roots, or zeroes, of the equation. The roots of a quadratic function are the same as its zeroes. They are where the graph crosses the x-axis, or simply put, where y = 0. A quadratic function can have 0, 1, or 2 roots.

Example 1:

y = 4x2 - 6x + 7

This problem cannot be factored and there is no easy way to solve for x when y = 0. So we must use the quadratic formula.

Step 1: First we find a, b, and c.

y = 4x2 - 6x + 7

a      b      c

This equation is already written in the form of y = ax2 + bx + c so we have a = 4, b = -6, and c = 7.

Step 2 : Now we substitute these values into the formula and use the order of operations to simplify

 $x=\frac{-b±\sqrt{{b}^{2}-4ac}}{2a}$ Quadratic Formula $x=\frac{-\left(-6\right)±\sqrt{{\left(-6\right)}^{2}-4\left(4\right)\left(7\right)}}{2\left(4\right)}$ Substitute 4,-6, and 7 for a, b, and c, respectively. $x=\frac{6±\sqrt{{\left(-6\right)}^{2}-4\left(4\right)\left(7\right)}}{2\left(4\right)}$ -(-6) can be simplified to 6. $x=\frac{6±\sqrt{36-4\left(4\right)\left(7\right)}}{2\left(4\right)}$ (-6)2 = (-6)(-6) = 36. $x=\frac{6±\sqrt{36-112}}{8}$ 4(4)(7) is equal to 112 and multiplying 2 and 4 equals 8. $x=\frac{6±\sqrt{-76}}{8}$ 36 – 112 = -76 which is negative. Square roots of negative numbers are not possible in the set of real numbers, so we have no solution.

Example 2:

y = - 16x + x2 -7

Rewriting this so that it is in the form of y = ax2 + bx + c , we get

y = x2 – 16x - 7                        We get a = 1, b = -16, c = -7.

a      b      c

 $x=\frac{-\left(-16\right)±\sqrt{{\left(-16\right)}^{2}-4\left(1\right)\left(-7\right)}}{2\left(1\right)}$ Substitute 1,-16, and -7 into the formula. $x=\frac{16±\sqrt{{\left(-16\right)}^{2}-4\left(1\right)\left(-7\right)}}{2\left(1\right)}$ -(-16) simplifies to 16. $x=\frac{16±\sqrt{256-4\left(1\right)\left(-7\right)}}{2\left(1\right)}$ (-16)2 = (-16)(-16) = 256. $x=\frac{16±\sqrt{256+28}}{2}$ Multiplying 4(1)(-7) equals 28 and 2 (1)= 2. $x=\frac{16±\sqrt{284}}{2}$ 256+28 = 284 $x=\frac{16±2\sqrt{71}}{2}$ Simplify the radical as much as possible

 $x=8±\sqrt{71}$ Divide numerator and denominator by 2 to simplify The two roots are $x=8+\sqrt{71}$ and $x=8-\sqrt{71}$