Solve 78x^2=252-85x

$$78 x ^ { 2 } = 252 - 85 x$$

Solve for x

$$x = -\frac{63}{26} = -2\frac{11}{26} \approx -2.423076923$$

$$x = \frac{4}{3} = 1\frac{1}{3} \approx 1.333333333$$

Steps Using Factoring By Grouping

Subtract 252 from both sides.

$$78x^{2}-252=-85x$$

Add 85x to both sides.

$$78x^{2}-252+85x=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$78x^{2}+85x-252=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 78x^{2}+ax+bx-252. To find a and b, set up a system to be solved.

$$a+b=85 ab=78\left(-252\right)=-19656$$

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 -19656.

$$-1,19656 -2,9828 -3,6552 -4,4914 -6,3276 -7,2808 -8,2457 -9,2184 -12,1638 -13,1512 -14,1404 -18,1092 -21,936 -24,819 -26,756 -27,728 -28,702 -36,546 -39,504 -42,468 -52,378 -54,364 -56,351 -63,312 -72,273 -78,252 -84,234 -91,216 -104,189 -108,182 -117,168 -126,156$$

Calculate the sum for each pair.

$$-1+19656=19655 -2+9828=9826 -3+6552=6549 -4+4914=4910 -6+3276=3270 -7+2808=2801 -8+2457=2449 -9+2184=2175 -12+1638=1626 -13+1512=1499 -14+1404=1390 -18+1092=1074 -21+936=915 -24+819=795 -26+756=730 -27+728=701 -28+702=674 -36+546=510 -39+504=465 -42+468=426 -52+378=326 -54+364=310 -56+351=295 -63+312=249 -72+273=201 -78+252=174 -84+234=150 -91+216=125 -104+189=85 -108+182=74 -117+168=51 -126+156=30$$

The solution is the pair that gives sum 85.

$$a=-104 b=189$$

Rewrite 78x^{2}+85x-252 as \left(78x^{2}-104x\right)+\left(189x-252\right).

$$\left(78x^{2}-104x\right)+\left(189x-252\right)$$

Factor out 26x in the first and 63 in the second group.

$$26x\left(3x-4\right)+63\left(3x-4\right)$$

Factor out common term 3x-4 by using distributive property.

$$\left(3x-4\right)\left(26x+63\right)$$

To find equation solutions, solve 3x-4=0 and 26x+63=0.

$$x=\frac{4}{3} x=-\frac{63}{26}$$

Steps Using the Quadratic Formula

Subtract 252 from both sides.

$$78x^{2}-252=-85x$$

Add 85x to both sides.

$$78x^{2}-252+85x=0$$

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.

$$78x^{2}+85x-252=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 78 for a, 85 for b, and -252 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-85±\sqrt{85^{2}-4\times 78\left(-252\right)}}{2\times 78}$$

Square 85.

$$x=\frac{-85±\sqrt{7225-4\times 78\left(-252\right)}}{2\times 78}$$

Multiply -4 times 78.

$$x=\frac{-85±\sqrt{7225-312\left(-252\right)}}{2\times 78}$$

Multiply -312 times -252.

$$x=\frac{-85±\sqrt{7225+78624}}{2\times 78}$$

Add 7225 to 78624.

$$x=\frac{-85±\sqrt{85849}}{2\times 78}$$

Take the square root of 85849.

$$x=\frac{-85±293}{2\times 78}$$

Multiply 2 times 78.

$$x=\frac{-85±293}{156}$$

Now solve the equation x=\frac{-85±293}{156} when ± is plus. Add -85 to 293.

$$x=\frac{208}{156}$$

Reduce the fraction \frac{208}{156} to lowest terms by extracting and canceling out 52.

$$x=\frac{4}{3}$$

Now solve the equation x=\frac{-85±293}{156} when ± is minus. Subtract 293 from -85.

$$x=-\frac{378}{156}$$

Reduce the fraction \frac{-378}{156} to lowest terms by extracting and canceling out 6.

$$x=-\frac{63}{26}$$

The equation is now solved.

$$x=\frac{4}{3} x=-\frac{63}{26}$$

Steps for Completing the Square

Add 85x to both sides.

$$78x^{2}+85x=252$$

Divide both sides by 78.

$$\frac{78x^{2}+85x}{78}=\frac{252}{78}$$

Dividing by 78 undoes the multiplication by 78.

$$x^{2}+\frac{85}{78}x=\frac{252}{78}$$

Reduce the fraction \frac{252}{78} to lowest terms by extracting and canceling out 6.

$$x^{2}+\frac{85}{78}x=\frac{42}{13}$$

Divide \frac{85}{78}, the coefficient of the x term, by 2 to get \frac{85}{156}. Then add the square of \frac{85}{156} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{85}{78}x+\left(\frac{85}{156}\right)^{2}=\frac{42}{13}+\left(\frac{85}{156}\right)^{2}$$

Square \frac{85}{156} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{85}{78}x+\frac{7225}{24336}=\frac{42}{13}+\frac{7225}{24336}$$

Add \frac{42}{13} to \frac{7225}{24336} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{85}{78}x+\frac{7225}{24336}=\frac{85849}{24336}$$

Factor x^{2}+\frac{85}{78}x+\frac{7225}{24336}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{85}{156}\right)^{2}=\frac{85849}{24336}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{85}{156}\right)^{2}}=\sqrt{\frac{85849}{24336}}$$

Simplify.

$$x+\frac{85}{156}=\frac{293}{156} x+\frac{85}{156}=-\frac{293}{156}$$

Subtract \frac{85}{156} from both sides of the equation.

$$x=\frac{4}{3} x=-\frac{63}{26}$$