Solve -3x-6=4x^2+169+52x

$$- 3 x - 6 = 4 x ^ { 2 } + 169 + 52 x$$

Solve for x

$$x = -\frac{35}{4} = -8\frac{3}{4} = -8.75$$

$$x=-5$$

Steps Using Factoring By Grouping

Subtract 4x^{2} from both sides.

$$-3x-6-4x^{2}=169+52x$$

Subtract 169 from both sides.

$$-3x-6-4x^{2}-169=52x$$

Subtract 169 from -6 to get -175.

$$-3x-175-4x^{2}=52x$$

Subtract 52x from both sides.

$$-3x-175-4x^{2}-52x=0$$

Combine -3x and -52x to get -55x.

$$-55x-175-4x^{2}=0$$

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

$$-4x^{2}-55x-175=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -4x^{2}+ax+bx-175. To find a and b, set up a system to be solved.

$$a+b=-55 ab=-4\left(-175\right)=700$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 700.

$$-1,-700 -2,-350 -4,-175 -5,-140 -7,-100 -10,-70 -14,-50 -20,-35 -25,-28$$

Calculate the sum for each pair.

$$-1-700=-701 -2-350=-352 -4-175=-179 -5-140=-145 -7-100=-107 -10-70=-80 -14-50=-64 -20-35=-55 -25-28=-53$$

The solution is the pair that gives sum -55.

$$a=-20 b=-35$$

Rewrite -4x^{2}-55x-175 as \left(-4x^{2}-20x\right)+\left(-35x-175\right).

$$\left(-4x^{2}-20x\right)+\left(-35x-175\right)$$

Factor out 4x in the first and 35 in the second group.

$$4x\left(-x-5\right)+35\left(-x-5\right)$$

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

$$\left(-x-5\right)\left(4x+35\right)$$

To find equation solutions, solve -x-5=0 and 4x+35=0.

$$x=-5 x=-\frac{35}{4}$$

Steps Using the Quadratic Formula

Subtract 4x^{2} from both sides.

$$-3x-6-4x^{2}=169+52x$$

Subtract 169 from both sides.

$$-3x-6-4x^{2}-169=52x$$

Subtract 169 from -6 to get -175.

$$-3x-175-4x^{2}=52x$$

Subtract 52x from both sides.

$$-3x-175-4x^{2}-52x=0$$

Combine -3x and -52x to get -55x.

$$-55x-175-4x^{2}=0$$

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.

$$-4x^{2}-55x-175=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, -55 for b, and -175 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-55\right)±\sqrt{\left(-55\right)^{2}-4\left(-4\right)\left(-175\right)}}{2\left(-4\right)}$$

Square -55.

$$x=\frac{-\left(-55\right)±\sqrt{3025-4\left(-4\right)\left(-175\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-\left(-55\right)±\sqrt{3025+16\left(-175\right)}}{2\left(-4\right)}$$

Multiply 16 times -175.

$$x=\frac{-\left(-55\right)±\sqrt{3025-2800}}{2\left(-4\right)}$$

Add 3025 to -2800.

$$x=\frac{-\left(-55\right)±\sqrt{225}}{2\left(-4\right)}$$

Take the square root of 225.

$$x=\frac{-\left(-55\right)±15}{2\left(-4\right)}$$

The opposite of -55 is 55.

$$x=\frac{55±15}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{55±15}{-8}$$

Now solve the equation x=\frac{55±15}{-8} when ± is plus. Add 55 to 15.

$$x=\frac{70}{-8}$$

Reduce the fraction \frac{70}{-8} to lowest terms by extracting and canceling out 2.

$$x=-\frac{35}{4}$$

Now solve the equation x=\frac{55±15}{-8} when ± is minus. Subtract 15 from 55.

$$x=\frac{40}{-8}$$

Divide 40 by -8.

$$x=-5$$

The equation is now solved.

$$x=-\frac{35}{4} x=-5$$

Steps for Completing the Square

Subtract 4x^{2} from both sides.

$$-3x-6-4x^{2}=169+52x$$

Subtract 52x from both sides.

$$-3x-6-4x^{2}-52x=169$$

Combine -3x and -52x to get -55x.

$$-55x-6-4x^{2}=169$$

Add 6 to both sides.

$$-55x-4x^{2}=169+6$$

Add 169 and 6 to get 175.

$$-55x-4x^{2}=175$$

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.

$$-4x^{2}-55x=175$$

Divide both sides by -4.

$$\frac{-4x^{2}-55x}{-4}=\frac{175}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\left(-\frac{55}{-4}\right)x=\frac{175}{-4}$$

Divide -55 by -4.

$$x^{2}+\frac{55}{4}x=\frac{175}{-4}$$

Divide 175 by -4.

$$x^{2}+\frac{55}{4}x=-\frac{175}{4}$$

Divide \frac{55}{4}, the coefficient of the x term, by 2 to get \frac{55}{8}. Then add the square of \frac{55}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{55}{4}x+\left(\frac{55}{8}\right)^{2}=-\frac{175}{4}+\left(\frac{55}{8}\right)^{2}$$

Square \frac{55}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{55}{4}x+\frac{3025}{64}=-\frac{175}{4}+\frac{3025}{64}$$

Add -\frac{175}{4} to \frac{3025}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{55}{4}x+\frac{3025}{64}=\frac{225}{64}$$

Factor x^{2}+\frac{55}{4}x+\frac{3025}{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}.

$$\left(x+\frac{55}{8}\right)^{2}=\frac{225}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{55}{8}\right)^{2}}=\sqrt{\frac{225}{64}}$$

Simplify.

$$x+\frac{55}{8}=\frac{15}{8} x+\frac{55}{8}=-\frac{15}{8}$$

Subtract \frac{55}{8} from both sides of the equation.

$$x=-5 x=-\frac{35}{4}$$