Solve x^2+14x+49=0

$${ x }^{ 2 } +14x+49=0$$

Solve for x

$$x=-7$$

Steps Using Factoring

To solve the equation, factor x^{2}+14x+49 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=14 ab=49$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 49.

$$1,49 7,7$$

Calculate the sum for each pair.

$$1+49=50 7+7=14$$

The solution is the pair that gives sum 14.

$$a=7 b=7$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x+7\right)\left(x+7\right)$$

Rewrite as a binomial square.

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

To find equation solution, solve x+7=0.

$$x=-7$$

Steps Using Factoring By Grouping

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+49. To find a and b, set up a system to be solved.

$$a+b=14 ab=1\times 49=49$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 49.

$$1,49 7,7$$

Calculate the sum for each pair.

$$1+49=50 7+7=14$$

The solution is the pair that gives sum 14.

$$a=7 b=7$$

Rewrite x^{2}+14x+49 as \left(x^{2}+7x\right)+\left(7x+49\right).

$$\left(x^{2}+7x\right)+\left(7x+49\right)$$

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

$$x\left(x+7\right)+7\left(x+7\right)$$

Factor out common term x+7 by using distributive property.

$$\left(x+7\right)\left(x+7\right)$$

Rewrite as a binomial square.

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

To find equation solution, solve x+7=0.

$$x=-7$$

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.

$$x^{2}+14x+49=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 14 for b, and 49 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\times 49}}{2}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\times 49}}{2}$$

Multiply -4 times 49.

$$x=\frac{-14±\sqrt{196-196}}{2}$$

Add 196 to -196.

$$x=\frac{-14±\sqrt{0}}{2}$$

Take the square root of 0.

$$x=-\frac{14}{2}$$

Divide -14 by 2.

$$x=-7$$

Steps for Completing the Square

Factor x^{2}+14x+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+7\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+7\right)^{2}}=\sqrt{0}$$

Simplify.

$$x+7=0 x+7=0$$

Subtract 7 from both sides of the equation.

$$x=-7 x=-7$$

The equation is now solved. Solutions are the same.

$$x=-7$$