Solve 12x-7x^2+64=0

$$12 x - 7 x ^ { 2 } + 64 = 0$$

Solve for x

$$x = -\frac{16}{7} = -2\frac{2}{7} \approx -2.285714286$$

$$x=4$$

Steps Using Factoring By Grouping

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-7x^{2}+12x+64=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -7x^{2}+ax+bx+64. To find a and b, set up a system to be solved.

$$a+b=12 ab=-7\times 64=-448$$

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 -448.

$$-1,448 -2,224 -4,112 -7,64 -8,56 -14,32 -16,28$$

Calculate the sum for each pair.

$$-1+448=447 -2+224=222 -4+112=108 -7+64=57 -8+56=48 -14+32=18 -16+28=12$$

The solution is the pair that gives sum 12.

$$a=28 b=-16$$

Rewrite -7x^{2}+12x+64 as \left(-7x^{2}+28x\right)+\left(-16x+64\right).

$$\left(-7x^{2}+28x\right)+\left(-16x+64\right)$$

Factor out 7x in the first and 16 in the second group.

$$7x\left(-x+4\right)+16\left(-x+4\right)$$

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

$$\left(-x+4\right)\left(7x+16\right)$$

To find equation solutions, solve -x+4=0 and 7x+16=0.

$$x=4 x=-\frac{16}{7}$$

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.

$$-7x^{2}+12x+64=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, 12 for b, and 64 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\left(-7\right)\times 64}}{2\left(-7\right)}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\left(-7\right)\times 64}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-12±\sqrt{144+28\times 64}}{2\left(-7\right)}$$

Multiply 28 times 64.

$$x=\frac{-12±\sqrt{144+1792}}{2\left(-7\right)}$$

Add 144 to 1792.

$$x=\frac{-12±\sqrt{1936}}{2\left(-7\right)}$$

Take the square root of 1936.

$$x=\frac{-12±44}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{-12±44}{-14}$$

Now solve the equation x=\frac{-12±44}{-14} when ± is plus. Add -12 to 44.

$$x=\frac{32}{-14}$$

Reduce the fraction \frac{32}{-14} to lowest terms by extracting and canceling out 2.

$$x=-\frac{16}{7}$$

Now solve the equation x=\frac{-12±44}{-14} when ± is minus. Subtract 44 from -12.

$$x=-\frac{56}{-14}$$

Divide -56 by -14.

$$x=4$$

The equation is now solved.

$$x=-\frac{16}{7} x=4$$

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.

$$-7x^{2}+12x+64=0$$

Subtract 64 from both sides of the equation.

$$-7x^{2}+12x+64-64=-64$$

Subtracting 64 from itself leaves 0.

$$-7x^{2}+12x=-64$$

Divide both sides by -7.

$$\frac{-7x^{2}+12x}{-7}=-\frac{64}{-7}$$

Dividing by -7 undoes the multiplication by -7.

$$x^{2}+\frac{12}{-7}x=-\frac{64}{-7}$$

Divide 12 by -7.

$$x^{2}-\frac{12}{7}x=-\frac{64}{-7}$$

Divide -64 by -7.

$$x^{2}-\frac{12}{7}x=\frac{64}{7}$$

Divide -\frac{12}{7}, the coefficient of the x term, by 2 to get -\frac{6}{7}. Then add the square of -\frac{6}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{12}{7}x+\left(-\frac{6}{7}\right)^{2}=\frac{64}{7}+\left(-\frac{6}{7}\right)^{2}$$

Square -\frac{6}{7} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{12}{7}x+\frac{36}{49}=\frac{64}{7}+\frac{36}{49}$$

Add \frac{64}{7} to \frac{36}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{12}{7}x+\frac{36}{49}=\frac{484}{49}$$

Factor x^{2}-\frac{12}{7}x+\frac{36}{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}.

$$\left(x-\frac{6}{7}\right)^{2}=\frac{484}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{6}{7}\right)^{2}}=\sqrt{\frac{484}{49}}$$

Simplify.

$$x-\frac{6}{7}=\frac{22}{7} x-\frac{6}{7}=-\frac{22}{7}$$

Add \frac{6}{7} to both sides of the equation.

$$x=4 x=-\frac{16}{7}$$