Solve 8x^2-35=-18x

$$8 x ^ { 2 } - 35 = - 18 x$$

Solve for x

$$x = -\frac{7}{2} = -3\frac{1}{2} = -3.5$$

$$x = \frac{5}{4} = 1\frac{1}{4} = 1.25$$

Steps Using Factoring By Grouping

Add 18x to both sides.

$$8x^{2}-35+18x=0$$

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

$$8x^{2}+18x-35=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 8x^{2}+ax+bx-35. To find a and b, set up a system to be solved.

$$a+b=18 ab=8\left(-35\right)=-280$$

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 -280.

$$-1,280 -2,140 -4,70 -5,56 -7,40 -8,35 -10,28 -14,20$$

Calculate the sum for each pair.

$$-1+280=279 -2+140=138 -4+70=66 -5+56=51 -7+40=33 -8+35=27 -10+28=18 -14+20=6$$

The solution is the pair that gives sum 18.

$$a=-10 b=28$$

Rewrite 8x^{2}+18x-35 as \left(8x^{2}-10x\right)+\left(28x-35\right).

$$\left(8x^{2}-10x\right)+\left(28x-35\right)$$

Factor out 2x in the first and 7 in the second group.

$$2x\left(4x-5\right)+7\left(4x-5\right)$$

Factor out common term 4x-5 by using distributive property.

$$\left(4x-5\right)\left(2x+7\right)$$

To find equation solutions, solve 4x-5=0 and 2x+7=0.

$$x=\frac{5}{4} x=-\frac{7}{2}$$

Steps Using the Quadratic Formula

Add 18x to both sides.

$$8x^{2}-35+18x=0$$

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.

$$8x^{2}+18x-35=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, 18 for b, and -35 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18±\sqrt{18^{2}-4\times 8\left(-35\right)}}{2\times 8}$$

Square 18.

$$x=\frac{-18±\sqrt{324-4\times 8\left(-35\right)}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-18±\sqrt{324-32\left(-35\right)}}{2\times 8}$$

Multiply -32 times -35.

$$x=\frac{-18±\sqrt{324+1120}}{2\times 8}$$

Add 324 to 1120.

$$x=\frac{-18±\sqrt{1444}}{2\times 8}$$

Take the square root of 1444.

$$x=\frac{-18±38}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{-18±38}{16}$$

Now solve the equation x=\frac{-18±38}{16} when ± is plus. Add -18 to 38.

$$x=\frac{20}{16}$$

Reduce the fraction \frac{20}{16} to lowest terms by extracting and canceling out 4.

$$x=\frac{5}{4}$$

Now solve the equation x=\frac{-18±38}{16} when ± is minus. Subtract 38 from -18.

$$x=-\frac{56}{16}$$

Reduce the fraction \frac{-56}{16} to lowest terms by extracting and canceling out 8.

$$x=-\frac{7}{2}$$

The equation is now solved.

$$x=\frac{5}{4} x=-\frac{7}{2}$$

Steps for Completing the Square

Add 18x to both sides.

$$8x^{2}-35+18x=0$$

Add 35 to both sides. Anything plus zero gives itself.

$$8x^{2}+18x=35$$

Divide both sides by 8.

$$\frac{8x^{2}+18x}{8}=\frac{35}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$x^{2}+\frac{18}{8}x=\frac{35}{8}$$

Reduce the fraction \frac{18}{8} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{9}{4}x=\frac{35}{8}$$

Divide \frac{9}{4}, the coefficient of the x term, by 2 to get \frac{9}{8}. Then add the square of \frac{9}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{9}{4}x+\left(\frac{9}{8}\right)^{2}=\frac{35}{8}+\left(\frac{9}{8}\right)^{2}$$

Square \frac{9}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{9}{4}x+\frac{81}{64}=\frac{35}{8}+\frac{81}{64}$$

Add \frac{35}{8} to \frac{81}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{9}{4}x+\frac{81}{64}=\frac{361}{64}$$

Factor x^{2}+\frac{9}{4}x+\frac{81}{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}.

$$\left(x+\frac{9}{8}\right)^{2}=\frac{361}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{9}{8}\right)^{2}}=\sqrt{\frac{361}{64}}$$

Simplify.

$$x+\frac{9}{8}=\frac{19}{8} x+\frac{9}{8}=-\frac{19}{8}$$

Subtract \frac{9}{8} from both sides of the equation.

$$x=\frac{5}{4} x=-\frac{7}{2}$$