# Quadratic Formula

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 = ax^{2} + bx + c, the quadratic formula below

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 = 4x^{2} - 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 = 4x^{2}- 6x + 7

a b c

This equation is already written in the form of y = ax^{2} + 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

$x=\frac{-b\pm \sqrt{{b}^{2}-4ac}}{2a}$ | Quadratic Formula |

$x=\frac{-(-6)\pm \sqrt{{(-6)}^{2}-4\left(4\right)\left(7\right)}}{2\left(4\right)}$ | Substitute 4,-6, and 7 for a, b, and c, respectively. |

$x=\frac{6\pm \sqrt{{(-6)}^{2}-4\left(4\right)\left(7\right)}}{2\left(4\right)}$ | -(-6) can be simplified to 6. |

$x=\frac{6\pm \sqrt{36-4\left(4\right)\left(7\right)}}{2\left(4\right)}$ | (-6)^{2} = (-6)(-6) = 36. |

$x=\frac{6\pm \sqrt{36-112}}{8}$ | 4(4)(7) is equal to 112 and multiplying 2 and 4 equals 8. |

$x=\frac{6\pm \sqrt{-76}}{8}$ | 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. |

Answer : no roots

Example 2:

y = - 16x + x2 -7

Rewriting this so that it is in the form of y = ax^{2} + bx + c , we get

y = x^{2} – 16x - 7 We get a = 1, b = -16, c = -7.

a b c

$x=\frac{-(-16)\pm \sqrt{{(-16)}^{2}-4\left(1\right)(-7)}}{2\left(1\right)}$ | Substitute 1,-16, and -7 into the formula. |

$x=\frac{16\pm \sqrt{{(-16)}^{2}-4\left(1\right)(-7)}}{2\left(1\right)}$ | -(-16) simplifies to 16. |

$x=\frac{16\pm \sqrt{256-4\left(1\right)(-7)}}{2\left(1\right)}$ | (-16)^{2} = (-16)(-16) = 256. |

$x=\frac{16\pm \sqrt{256+28}}{2}$ | Multiplying 4(1)(-7) equals 28 and 2 (1)= 2. |

$x=\frac{16\pm \sqrt{284}}{2}$ | 256+28 = 284 |

$x=\frac{16\pm 2\sqrt{71}}{2}$ | Simplify the radical as much as possible |

Answer:

$x=8\pm \sqrt{71}$ | Divide numerator and denominator by 2 to simplify |

The two roots are $x=8+\sqrt{71}$ and $x=8-\sqrt{71}$ |

Related Links:Quadratic Functions Solving Quadratic Equations Quiz Quadratic Function Standard Form Quadratic Function Vertex Form |