Solve 1/4x^2+31/2x+12=0

$$\frac { 1 } { 4 } x ^ { 2 } + 3 \frac { 1 } { 2 } x + 12 = 0$$

Solve for x

$$x=-8$$

$$x=-6$$

Steps Using Factoring

Multiply both sides of the equation by 4, the least common multiple of 4,2.

$$1x^{2}+2\left(3\times 2+1\right)x+48=0$$

Multiply 3 and 2 to get 6.

$$1x^{2}+2\left(6+1\right)x+48=0$$

Add 6 and 1 to get 7.

$$1x^{2}+2\times 7x+48=0$$

Multiply 2 and 7 to get 14.

$$1x^{2}+14x+48=0$$

Reorder the terms.

$$x^{2}+14x+48=0$$

To solve the equation, factor x^{2}+14x+48 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.

$$a+b=14 ab=48$$

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 48.

$$1,48 2,24 3,16 4,12 6,8$$

Calculate the sum for each pair.

$$1+48=49 2+24=26 3+16=19 4+12=16 6+8=14$$

The solution is the pair that gives sum 14.

$$a=6 b=8$$

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

$$\left(x+6\right)\left(x+8\right)$$

To find equation solutions, solve x+6=0 and x+8=0.

$$x=-6 x=-8$$

Steps Using Factoring By Grouping

Multiply both sides of the equation by 4, the least common multiple of 4,2.

$$1x^{2}+2\left(3\times 2+1\right)x+48=0$$

Multiply 3 and 2 to get 6.

$$1x^{2}+2\left(6+1\right)x+48=0$$

Add 6 and 1 to get 7.

$$1x^{2}+2\times 7x+48=0$$

Multiply 2 and 7 to get 14.

$$1x^{2}+14x+48=0$$

Reorder the terms.

$$x^{2}+14x+48=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+48. To find a and b, set up a system to be solved.

$$a+b=14 ab=1\times 48=48$$

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 48.

$$1,48 2,24 3,16 4,12 6,8$$

Calculate the sum for each pair.

$$1+48=49 2+24=26 3+16=19 4+12=16 6+8=14$$

The solution is the pair that gives sum 14.

$$a=6 b=8$$

Rewrite x^{2}+14x+48 as \left(x^{2}+6x\right)+\left(8x+48\right).

$$\left(x^{2}+6x\right)+\left(8x+48\right)$$

Factor out x in the first and 8 in the second group.

$$x\left(x+6\right)+8\left(x+6\right)$$

Factor out common term x+6 by using distributive property.

$$\left(x+6\right)\left(x+8\right)$$

To find equation solutions, solve x+6=0 and x+8=0.

$$x=-6 x=-8$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 4, the least common multiple of 4,2.

$$1x^{2}+2\left(3\times 2+1\right)x+48=0$$

Multiply 3 and 2 to get 6.

$$1x^{2}+2\left(6+1\right)x+48=0$$

Add 6 and 1 to get 7.

$$1x^{2}+2\times 7x+48=0$$

Multiply 2 and 7 to get 14.

$$1x^{2}+14x+48=0$$

Reorder the terms.

$$x^{2}+14x+48=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 14 for b, and 48 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\times 48}}{2}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\times 48}}{2}$$

Multiply -4 times 48.

$$x=\frac{-14±\sqrt{196-192}}{2}$$

Add 196 to -192.

$$x=\frac{-14±\sqrt{4}}{2}$$

Take the square root of 4.

$$x=\frac{-14±2}{2}$$

Now solve the equation x=\frac{-14±2}{2} when ± is plus. Add -14 to 2.

$$x=-\frac{12}{2}$$

Divide -12 by 2.

$$x=-6$$

Now solve the equation x=\frac{-14±2}{2} when ± is minus. Subtract 2 from -14.

$$x=-\frac{16}{2}$$

Divide -16 by 2.

$$x=-8$$

The equation is now solved.

$$x=-6 x=-8$$

Steps for Completing the Square

Multiply both sides of the equation by 4, the least common multiple of 4,2.

$$1x^{2}+2\left(3\times 2+1\right)x+48=0$$

Multiply 3 and 2 to get 6.

$$1x^{2}+2\left(6+1\right)x+48=0$$

Add 6 and 1 to get 7.

$$1x^{2}+2\times 7x+48=0$$

Multiply 2 and 7 to get 14.

$$1x^{2}+14x+48=0$$

Subtract 48 from both sides. Anything subtracted from zero gives its negation.

$$1x^{2}+14x=-48$$

Reorder the terms.

$$x^{2}+14x=-48$$

Divide 14, the coefficient of the x term, by 2 to get 7. Then add the square of 7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+14x+7^{2}=-48+7^{2}$$

Square 7.

$$x^{2}+14x+49=-48+49$$

Add -48 to 49.

$$x^{2}+14x+49=1$$

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}.

$$\left(x+7\right)^{2}=1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+7\right)^{2}}=\sqrt{1}$$

Simplify.

$$x+7=1 x+7=-1$$

Subtract 7 from both sides of the equation.

$$x=-6 x=-8$$