Solve 4x^2+28x+49=169

$$4 x ^ { 2 } + 28 x + 49 = 169$$

Solve for x

$$x=-10$$

$$x=3$$

Steps Using Factoring By Grouping

Subtract 169 from both sides.

$$4x^{2}+28x+49-169=0$$

Subtract 169 from 49 to get -120.

$$4x^{2}+28x-120=0$$

Divide both sides by 4.

$$x^{2}+7x-30=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-30. To find a and b, set up a system to be solved.

$$a+b=7 ab=1\left(-30\right)=-30$$

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 -30.

$$-1,30 -2,15 -3,10 -5,6$$

Calculate the sum for each pair.

$$-1+30=29 -2+15=13 -3+10=7 -5+6=1$$

The solution is the pair that gives sum 7.

$$a=-3 b=10$$

Rewrite x^{2}+7x-30 as \left(x^{2}-3x\right)+\left(10x-30\right).

$$\left(x^{2}-3x\right)+\left(10x-30\right)$$

Factor out x in the first and 10 in the second group.

$$x\left(x-3\right)+10\left(x-3\right)$$

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

$$\left(x-3\right)\left(x+10\right)$$

To find equation solutions, solve x-3=0 and x+10=0.

$$x=3 x=-10$$

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.

$$4x^{2}+28x+49=169$$

Subtract 169 from both sides of the equation.

$$4x^{2}+28x+49-169=169-169$$

Subtracting 169 from itself leaves 0.

$$4x^{2}+28x+49-169=0$$

Subtract 169 from 49.

$$4x^{2}+28x-120=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 28 for b, and -120 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-28±\sqrt{28^{2}-4\times 4\left(-120\right)}}{2\times 4}$$

Square 28.

$$x=\frac{-28±\sqrt{784-4\times 4\left(-120\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-28±\sqrt{784-16\left(-120\right)}}{2\times 4}$$

Multiply -16 times -120.

$$x=\frac{-28±\sqrt{784+1920}}{2\times 4}$$

Add 784 to 1920.

$$x=\frac{-28±\sqrt{2704}}{2\times 4}$$

Take the square root of 2704.

$$x=\frac{-28±52}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-28±52}{8}$$

Now solve the equation x=\frac{-28±52}{8} when ± is plus. Add -28 to 52.

$$x=\frac{24}{8}$$

Divide 24 by 8.

$$x=3$$

Now solve the equation x=\frac{-28±52}{8} when ± is minus. Subtract 52 from -28.

$$x=-\frac{80}{8}$$

Divide -80 by 8.

$$x=-10$$

The equation is now solved.

$$x=3 x=-10$$

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.

$$4x^{2}+28x+49=169$$

Subtract 49 from both sides of the equation.

$$4x^{2}+28x+49-49=169-49$$

Subtracting 49 from itself leaves 0.

$$4x^{2}+28x=169-49$$

Subtract 49 from 169.

$$4x^{2}+28x=120$$

Divide both sides by 4.

$$\frac{4x^{2}+28x}{4}=\frac{120}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{28}{4}x=\frac{120}{4}$$

Divide 28 by 4.

$$x^{2}+7x=\frac{120}{4}$$

Divide 120 by 4.

$$x^{2}+7x=30$$

Divide 7, the coefficient of the x term, by 2 to get \frac{7}{2}. Then add the square of \frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+7x+\left(\frac{7}{2}\right)^{2}=30+\left(\frac{7}{2}\right)^{2}$$

Square \frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+7x+\frac{49}{4}=30+\frac{49}{4}$$

Add 30 to \frac{49}{4}.

$$x^{2}+7x+\frac{49}{4}=\frac{169}{4}$$

Factor x^{2}+7x+\frac{49}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{7}{2}\right)^{2}=\frac{169}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{2}\right)^{2}}=\sqrt{\frac{169}{4}}$$

Simplify.

$$x+\frac{7}{2}=\frac{13}{2} x+\frac{7}{2}=-\frac{13}{2}$$

Subtract \frac{7}{2} from both sides of the equation.

$$x=3 x=-10$$