Quadratic Formula

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 = ax2 + bx + c, the quadratic formula below

x=-b±b2-4ac2a

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=-b±b2-4ac2aQuadratic Formula
x=-(-6)±(-6)2-4(4)(7)2(4)Substitute 4,-6, and 7 for a, b, and c, respectively.
x=6±(-6)2-4(4)(7)2(4)-(-6) can be simplified to 6.
x=6±36-4(4)(7)2(4)(-6)2 = (-6)(-6) = 36.
x=6±36-11284(4)(7) is equal to 112 and multiplying 2 and 4 equals 8.
x=6±-76836 – 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 = ax2 + bx + c , we get

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

       a      b      c

x=-(-16)±(-16)2-4(1)(-7)2(1)Substitute 1,-16, and -7 into the formula.
x=16±(-16)2-4(1)(-7)2(1)-(-16) simplifies to 16.
x=16±256-4(1)(-7)2(1)(-16)2 = (-16)(-16) = 256.
x=16±256+282Multiplying 4(1)(-7) equals 28 and 2 (1)= 2.
x=16±2842256+28 = 284
x=16±2712Simplify the radical as much as possible

Answer:

x=8±71Divide numerator and denominator by 2 to simplify
The two roots are x=8+71 and x=8-71
Related Links:
Quadratic Functions
Solving Quadratic Equations Quiz
Quadratic Function Standard Form
Quadratic Function Vertex Form






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