Solve a^2+14a+49=0

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

Solve for a

$$a=-7$$

Steps Using Factoring

To solve the equation, factor a^{2}+14a+49 using formula a^{2}+\left(a+b\right)a+ab=\left(a+a\right)\left(a+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(a+a\right)\left(a+b\right) using the obtained values.

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

Rewrite as a binomial square.

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

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

$$a=-7$$

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 a^{2}+aa+ba+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 a^{2}+14a+49 as \left(a^{2}+7a\right)+\left(7a+49\right).

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

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

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

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

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

Rewrite as a binomial square.

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

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

$$a=-7$$

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.

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

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}.

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

Square 14.

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

Multiply -4 times 49.

$$a=\frac{-14±\sqrt{196-196}}{2}$$

Add 196 to -196.

$$a=\frac{-14±\sqrt{0}}{2}$$

Take the square root of 0.

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

Divide -14 by 2.

$$a=-7$$

Steps for Completing the Square

Factor a^{2}+14a+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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$a+7=0 a+7=0$$

Subtract 7 from both sides of the equation.

$$a=-7 a=-7$$

The equation is now solved. Solutions are the same.

$$a=-7$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.

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

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = -14 rs = 49$$

Two numbers r and s sum up to -14 exactly when the average of the two numbers is \frac{1}{2}*-14 = -7. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = -7 - u s = -7 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 49

$$(-7 - u) (-7 + u) = 49$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$49 - u^2 = 49$$

Simplify the expression by subtracting 49 on both sides

$$-u^2 = 49-49 = 0$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = 0 u = 0$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r = s = -7$$