Solve (9x+8)^2=(5x-1)^2+4879

$$( 9 x + 8 ) ^ { 2 } = ( 5 x - 1 ) ^ { 2 } + 4879$$

Solve for x

$$x = -\frac{43}{4} = -10\frac{3}{4} = -10.75$$

$$x=8$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(9x+8\right)^{2}.

$$81x^{2}+144x+64=\left(5x-1\right)^{2}+4879$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5x-1\right)^{2}.

$$81x^{2}+144x+64=25x^{2}-10x+1+4879$$

Add 1 and 4879 to get 4880.

$$81x^{2}+144x+64=25x^{2}-10x+4880$$

Subtract 25x^{2} from both sides.

$$81x^{2}+144x+64-25x^{2}=-10x+4880$$

Combine 81x^{2} and -25x^{2} to get 56x^{2}.

$$56x^{2}+144x+64=-10x+4880$$

Add 10x to both sides.

$$56x^{2}+144x+64+10x=4880$$

Combine 144x and 10x to get 154x.

$$56x^{2}+154x+64=4880$$

Subtract 4880 from both sides.

$$56x^{2}+154x+64-4880=0$$

Subtract 4880 from 64 to get -4816.

$$56x^{2}+154x-4816=0$$

Divide both sides by 14.

$$4x^{2}+11x-344=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 4x^{2}+ax+bx-344. To find a and b, set up a system to be solved.

$$a+b=11 ab=4\left(-344\right)=-1376$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -1376.

$$-1,1376 -2,688 -4,344 -8,172 -16,86 -32,43$$

Calculate the sum for each pair.

$$-1+1376=1375 -2+688=686 -4+344=340 -8+172=164 -16+86=70 -32+43=11$$

The solution is the pair that gives sum 11.

$$a=-32 b=43$$

Rewrite 4x^{2}+11x-344 as \left(4x^{2}-32x\right)+\left(43x-344\right).

$$\left(4x^{2}-32x\right)+\left(43x-344\right)$$

Factor out 4x in the first and 43 in the second group.

$$4x\left(x-8\right)+43\left(x-8\right)$$

Factor out common term x-8 by using distributive property.

$$\left(x-8\right)\left(4x+43\right)$$

To find equation solutions, solve x-8=0 and 4x+43=0.

$$x=8 x=-\frac{43}{4}$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(9x+8\right)^{2}.

$$81x^{2}+144x+64=\left(5x-1\right)^{2}+4879$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5x-1\right)^{2}.

$$81x^{2}+144x+64=25x^{2}-10x+1+4879$$

Add 1 and 4879 to get 4880.

$$81x^{2}+144x+64=25x^{2}-10x+4880$$

Subtract 25x^{2} from both sides.

$$81x^{2}+144x+64-25x^{2}=-10x+4880$$

Combine 81x^{2} and -25x^{2} to get 56x^{2}.

$$56x^{2}+144x+64=-10x+4880$$

Add 10x to both sides.

$$56x^{2}+144x+64+10x=4880$$

Combine 144x and 10x to get 154x.

$$56x^{2}+154x+64=4880$$

Subtract 4880 from both sides.

$$56x^{2}+154x+64-4880=0$$

Subtract 4880 from 64 to get -4816.

$$56x^{2}+154x-4816=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 56 for a, 154 for b, and -4816 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-154±\sqrt{154^{2}-4\times 56\left(-4816\right)}}{2\times 56}$$

Square 154.

$$x=\frac{-154±\sqrt{23716-4\times 56\left(-4816\right)}}{2\times 56}$$

Multiply -4 times 56.

$$x=\frac{-154±\sqrt{23716-224\left(-4816\right)}}{2\times 56}$$

Multiply -224 times -4816.

$$x=\frac{-154±\sqrt{23716+1078784}}{2\times 56}$$

Add 23716 to 1078784.

$$x=\frac{-154±\sqrt{1102500}}{2\times 56}$$

Take the square root of 1102500.

$$x=\frac{-154±1050}{2\times 56}$$

Multiply 2 times 56.

$$x=\frac{-154±1050}{112}$$

Now solve the equation x=\frac{-154±1050}{112} when ± is plus. Add -154 to 1050.

$$x=\frac{896}{112}$$

Divide 896 by 112.

$$x=8$$

Now solve the equation x=\frac{-154±1050}{112} when ± is minus. Subtract 1050 from -154.

$$x=-\frac{1204}{112}$$

Reduce the fraction \frac{-1204}{112} to lowest terms by extracting and canceling out 28.

$$x=-\frac{43}{4}$$

The equation is now solved.

$$x=8 x=-\frac{43}{4}$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(9x+8\right)^{2}.

$$81x^{2}+144x+64=\left(5x-1\right)^{2}+4879$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5x-1\right)^{2}.

$$81x^{2}+144x+64=25x^{2}-10x+1+4879$$

Add 1 and 4879 to get 4880.

$$81x^{2}+144x+64=25x^{2}-10x+4880$$

Subtract 25x^{2} from both sides.

$$81x^{2}+144x+64-25x^{2}=-10x+4880$$

Combine 81x^{2} and -25x^{2} to get 56x^{2}.

$$56x^{2}+144x+64=-10x+4880$$

Add 10x to both sides.

$$56x^{2}+144x+64+10x=4880$$

Combine 144x and 10x to get 154x.

$$56x^{2}+154x+64=4880$$

Subtract 64 from both sides.

$$56x^{2}+154x=4880-64$$

Subtract 64 from 4880 to get 4816.

$$56x^{2}+154x=4816$$

Divide both sides by 56.

$$\frac{56x^{2}+154x}{56}=\frac{4816}{56}$$

Dividing by 56 undoes the multiplication by 56.

$$x^{2}+\frac{154}{56}x=\frac{4816}{56}$$

Reduce the fraction \frac{154}{56} to lowest terms by extracting and canceling out 14.

$$x^{2}+\frac{11}{4}x=\frac{4816}{56}$$

Divide 4816 by 56.

$$x^{2}+\frac{11}{4}x=86$$

Divide \frac{11}{4}, the coefficient of the x term, by 2 to get \frac{11}{8}. Then add the square of \frac{11}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{11}{4}x+\left(\frac{11}{8}\right)^{2}=86+\left(\frac{11}{8}\right)^{2}$$

Square \frac{11}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{11}{4}x+\frac{121}{64}=86+\frac{121}{64}$$

Add 86 to \frac{121}{64}.

$$x^{2}+\frac{11}{4}x+\frac{121}{64}=\frac{5625}{64}$$

Factor x^{2}+\frac{11}{4}x+\frac{121}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{11}{8}\right)^{2}=\frac{5625}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{11}{8}\right)^{2}}=\sqrt{\frac{5625}{64}}$$

Simplify.

$$x+\frac{11}{8}=\frac{75}{8} x+\frac{11}{8}=-\frac{75}{8}$$

Subtract \frac{11}{8} from both sides of the equation.

$$x=8 x=-\frac{43}{4}$$