Solve 0=-16x^2+100x+84

$$0=-16 { x }^{ 2 } +100x+84$$

Solve for x

$$x=7$$

$$x=-\frac{3}{4}=-0.75$$

Steps Using Factoring By Grouping

Swap sides so that all variable terms are on the left hand side.

$$-16x^{2}+100x+84=0$$

Divide both sides by 4.

$$-4x^{2}+25x+21=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -4x^{2}+ax+bx+21. To find a and b, set up a system to be solved.

$$a+b=25 ab=-4\times 21=-84$$

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 -84.

$$-1,84 -2,42 -3,28 -4,21 -6,14 -7,12$$

Calculate the sum for each pair.

$$-1+84=83 -2+42=40 -3+28=25 -4+21=17 -6+14=8 -7+12=5$$

The solution is the pair that gives sum 25.

$$a=28 b=-3$$

Rewrite -4x^{2}+25x+21 as \left(-4x^{2}+28x\right)+\left(-3x+21\right).

$$\left(-4x^{2}+28x\right)+\left(-3x+21\right)$$

Factor out 4x in the first and 3 in the second group.

$$4x\left(-x+7\right)+3\left(-x+7\right)$$

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

$$\left(-x+7\right)\left(4x+3\right)$$

To find equation solutions, solve -x+7=0 and 4x+3=0.

$$x=7 x=-\frac{3}{4}$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$-16x^{2}+100x+84=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 100 for b, and 84 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-100±\sqrt{100^{2}-4\left(-16\right)\times 84}}{2\left(-16\right)}$$

Square 100.

$$x=\frac{-100±\sqrt{10000-4\left(-16\right)\times 84}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$x=\frac{-100±\sqrt{10000+64\times 84}}{2\left(-16\right)}$$

Multiply 64 times 84.

$$x=\frac{-100±\sqrt{10000+5376}}{2\left(-16\right)}$$

Add 10000 to 5376.

$$x=\frac{-100±\sqrt{15376}}{2\left(-16\right)}$$

Take the square root of 15376.

$$x=\frac{-100±124}{2\left(-16\right)}$$

Multiply 2 times -16.

$$x=\frac{-100±124}{-32}$$

Now solve the equation x=\frac{-100±124}{-32} when ± is plus. Add -100 to 124.

$$x=\frac{24}{-32}$$

Reduce the fraction \frac{24}{-32} to lowest terms by extracting and canceling out 8.

$$x=-\frac{3}{4}$$

Now solve the equation x=\frac{-100±124}{-32} when ± is minus. Subtract 124 from -100.

$$x=-\frac{224}{-32}$$

Divide -224 by -32.

$$x=7$$

The equation is now solved.

$$x=-\frac{3}{4} x=7$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$-16x^{2}+100x+84=0$$

Subtract 84 from both sides. Anything subtracted from zero gives its negation.

$$-16x^{2}+100x=-84$$

Divide both sides by -16.

$$\frac{-16x^{2}+100x}{-16}=-\frac{84}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$x^{2}+\frac{100}{-16}x=-\frac{84}{-16}$$

Reduce the fraction \frac{100}{-16} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{25}{4}x=-\frac{84}{-16}$$

Reduce the fraction \frac{-84}{-16} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{25}{4}x=\frac{21}{4}$$

Divide -\frac{25}{4}, the coefficient of the x term, by 2 to get -\frac{25}{8}. Then add the square of -\frac{25}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{25}{4}x+\left(-\frac{25}{8}\right)^{2}=\frac{21}{4}+\left(-\frac{25}{8}\right)^{2}$$

Square -\frac{25}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{25}{4}x+\frac{625}{64}=\frac{21}{4}+\frac{625}{64}$$

Add \frac{21}{4} to \frac{625}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{25}{4}x+\frac{625}{64}=\frac{961}{64}$$

Factor x^{2}-\frac{25}{4}x+\frac{625}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{25}{8}\right)^{2}=\frac{961}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{25}{8}\right)^{2}}=\sqrt{\frac{961}{64}}$$

Simplify.

$$x-\frac{25}{8}=\frac{31}{8} x-\frac{25}{8}=-\frac{31}{8}$$

Add \frac{25}{8} to both sides of the equation.

$$x=7 x=-\frac{3}{4}$$