Solve 8x^2-313x-615=0

$$8 x ^ { 2 } - 313 x - 615 = 0$$

Solve for x

$$x = -\frac{15}{8} = -1\frac{7}{8} = -1.875$$

$$x=41$$

Steps Using Factoring By Grouping

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 8x^{2}+ax+bx-615. To find a and b, set up a system to be solved.

$$a+b=-313 ab=8\left(-615\right)=-4920$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -4920.

$$1,-4920 2,-2460 3,-1640 4,-1230 5,-984 6,-820 8,-615 10,-492 12,-410 15,-328 20,-246 24,-205 30,-164 40,-123 41,-120 60,-82$$

Calculate the sum for each pair.

$$1-4920=-4919 2-2460=-2458 3-1640=-1637 4-1230=-1226 5-984=-979 6-820=-814 8-615=-607 10-492=-482 12-410=-398 15-328=-313 20-246=-226 24-205=-181 30-164=-134 40-123=-83 41-120=-79 60-82=-22$$

The solution is the pair that gives sum -313.

$$a=-328 b=15$$

Rewrite 8x^{2}-313x-615 as \left(8x^{2}-328x\right)+\left(15x-615\right).

$$\left(8x^{2}-328x\right)+\left(15x-615\right)$$

Factor out 8x in the first and 15 in the second group.

$$8x\left(x-41\right)+15\left(x-41\right)$$

Factor out common term x-41 by using distributive property.

$$\left(x-41\right)\left(8x+15\right)$$

To find equation solutions, solve x-41=0 and 8x+15=0.

$$x=41 x=-\frac{15}{8}$$

Steps Using the Quadratic Formula

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$8x^{2}-313x-615=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, -313 for b, and -615 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-313\right)±\sqrt{\left(-313\right)^{2}-4\times 8\left(-615\right)}}{2\times 8}$$

Square -313.

$$x=\frac{-\left(-313\right)±\sqrt{97969-4\times 8\left(-615\right)}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-\left(-313\right)±\sqrt{97969-32\left(-615\right)}}{2\times 8}$$

Multiply -32 times -615.

$$x=\frac{-\left(-313\right)±\sqrt{97969+19680}}{2\times 8}$$

Add 97969 to 19680.

$$x=\frac{-\left(-313\right)±\sqrt{117649}}{2\times 8}$$

Take the square root of 117649.

$$x=\frac{-\left(-313\right)±343}{2\times 8}$$

The opposite of -313 is 313.

$$x=\frac{313±343}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{313±343}{16}$$

Now solve the equation x=\frac{313±343}{16} when ± is plus. Add 313 to 343.

$$x=\frac{656}{16}$$

Divide 656 by 16.

$$x=41$$

Now solve the equation x=\frac{313±343}{16} when ± is minus. Subtract 343 from 313.

$$x=-\frac{30}{16}$$

Reduce the fraction \frac{-30}{16} to lowest terms by extracting and canceling out 2.

$$x=-\frac{15}{8}$$

The equation is now solved.

$$x=41 x=-\frac{15}{8}$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$8x^{2}-313x-615=0$$

Add 615 to both sides of the equation.

$$8x^{2}-313x-615-\left(-615\right)=-\left(-615\right)$$

Subtracting -615 from itself leaves 0.

$$8x^{2}-313x=-\left(-615\right)$$

Subtract -615 from 0.

$$8x^{2}-313x=615$$

Divide both sides by 8.

$$\frac{8x^{2}-313x}{8}=\frac{615}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$x^{2}-\frac{313}{8}x=\frac{615}{8}$$

Divide -\frac{313}{8}, the coefficient of the x term, by 2 to get -\frac{313}{16}. Then add the square of -\frac{313}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{313}{8}x+\left(-\frac{313}{16}\right)^{2}=\frac{615}{8}+\left(-\frac{313}{16}\right)^{2}$$

Square -\frac{313}{16} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{313}{8}x+\frac{97969}{256}=\frac{615}{8}+\frac{97969}{256}$$

Add \frac{615}{8} to \frac{97969}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{313}{8}x+\frac{97969}{256}=\frac{117649}{256}$$

Factor x^{2}-\frac{313}{8}x+\frac{97969}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{313}{16}\right)^{2}=\frac{117649}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{313}{16}\right)^{2}}=\sqrt{\frac{117649}{256}}$$

Simplify.

$$x-\frac{313}{16}=\frac{343}{16} x-\frac{313}{16}=-\frac{343}{16}$$

Add \frac{313}{16} to both sides of the equation.

$$x=41 x=-\frac{15}{8}$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 8

$$x ^ 2 -\frac{313}{8}x -\frac{615}{8} = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = \frac{313}{8} rs = -\frac{615}{8}$$

Two numbers r and s sum up to \frac{313}{8} exactly when the average of the two numbers is \frac{1}{2}*\frac{313}{8} = \frac{313}{16}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = \frac{313}{16} - u s = \frac{313}{16} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{615}{8}

$$(\frac{313}{16} - u) (\frac{313}{16} + u) = -\frac{615}{8}$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$\frac{97969}{256} - u^2 = -\frac{615}{8}$$

Simplify the expression by subtracting \frac{97969}{256} on both sides

$$-u^2 = -\frac{615}{8}-\frac{97969}{256} = -\frac{117649}{256}$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = \frac{117649}{256} u = \pm\sqrt{\frac{117649}{256}} = \pm \frac{343}{16}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =\frac{313}{16} - \frac{343}{16} = -1.875 s = \frac{313}{16} + \frac{343}{16} = 41$$