Solve for x

x=-4

Graph

Copy

Copied to clipboard

a+b=8 ab=16

To solve the equation, factor x^{2}+8x+16 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.

1,16 2,8 4,4

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 16.

1+16=17 2+8=10 4+4=8

Calculate the sum for each pair.

a=4 b=4

The solution is the pair that gives sum 8.

\left(x+4\right)\left(x+4\right)

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

\left(x+4\right)^{2}

Rewrite as a binomial square.

x=-4

To find equation solution, solve x+4=0.

a+b=8 ab=1\times 16=16

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+16. To find a and b, set up a system to be solved.

1,16 2,8 4,4

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 16.

1+16=17 2+8=10 4+4=8

Calculate the sum for each pair.

a=4 b=4

The solution is the pair that gives sum 8.

\left(x^{2}+4x\right)+\left(4x+16\right)

Rewrite x^{2}+8x+16 as \left(x^{2}+4x\right)+\left(4x+16\right).

x\left(x+4\right)+4\left(x+4\right)

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

\left(x+4\right)\left(x+4\right)

Factor out common term x+4 by using distributive property.

\left(x+4\right)^{2}

Rewrite as a binomial square.

x=-4

To find equation solution, solve x+4=0.

x^{2}+8x+16=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.

x=\frac{-8±\sqrt{8^{2}-4\times 16}}{2}

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 8 for b, and 16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

x=\frac{-8±\sqrt{64-4\times 16}}{2}

Square 8.

x=\frac{-8±\sqrt{64-64}}{2}

Multiply -4 times 16.

x=\frac{-8±\sqrt{0}}{2}

Add 64 to -64.

x=-\frac{8}{2}

Take the square root of 0.

x=-4

Divide -8 by 2.

\left(x+4\right)^{2}=0

Factor x^{2}+8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

\sqrt{\left(x+4\right)^{2}}=\sqrt{0}

Take the square root of both sides of the equation.

x+4=0 x+4=0

Simplify.

x=-4 x=-4

Subtract 4 from both sides of the equation.

x=-4

The equation is now solved. Solutions are the same.

x ^ 2 +8x +16 = 0

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.

r + s = -8 rs = 16

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 = -4 - u s = -4 + u

Two numbers r and s sum up to -8 exactly when the average of the two numbers is \frac{1}{2}*-8 = -4. 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>

(-4 - u) (-4 + u) = 16

To solve for unknown quantity u, substitute these in the product equation rs = 16

16 - u^2 = 16

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

-u^2 = 16-16 = 0

Simplify the expression by subtracting 16 on both sides

u^2 = 0 u = 0

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

r = s = -4

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

Examples

Quadratic equation

{ x } ^ { 2 } - 4 x - 5 = 0

Trigonometry

4 \sin \theta \cos \theta = 2 \sin \theta

Linear equation

y = 3x + 4

Arithmetic

699 * 533

Matrix

\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]

Simultaneous equation

\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.

Differentiation

\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }

Integration

\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x

Limits

\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}