Solve -4x^2+19x=21

$$- 4 x ^ { 2 } + 19 x = 21$$

Solve for x

$$x=3$$

$$x = \frac{7}{4} = 1\frac{3}{4} = 1.75$$

Steps Using Factoring By Grouping

Subtract 21 from both sides.

$$-4x^{2}+19x-21=0$$

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.

$$a+b=19 ab=-4\left(-21\right)=84$$

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 84.

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

Calculate the sum for each pair.

$$1+84=85 2+42=44 3+28=31 4+21=25 6+14=20 7+12=19$$

The solution is the pair that gives sum 19.

$$a=12 b=7$$

Rewrite -4x^{2}+19x-21 as \left(-4x^{2}+12x\right)+\left(7x-21\right).

$$\left(-4x^{2}+12x\right)+\left(7x-21\right)$$

Factor out 4x in the first and -7 in the second group.

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

Factor out common term -x+3 by using distributive property.

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

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

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

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.

$$-4x^{2}+19x=21$$

Subtract 21 from both sides of the equation.

$$-4x^{2}+19x-21=21-21$$

Subtracting 21 from itself leaves 0.

$$-4x^{2}+19x-21=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 19 for b, and -21 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-19±\sqrt{19^{2}-4\left(-4\right)\left(-21\right)}}{2\left(-4\right)}$$

Square 19.

$$x=\frac{-19±\sqrt{361-4\left(-4\right)\left(-21\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-19±\sqrt{361+16\left(-21\right)}}{2\left(-4\right)}$$

Multiply 16 times -21.

$$x=\frac{-19±\sqrt{361-336}}{2\left(-4\right)}$$

Add 361 to -336.

$$x=\frac{-19±\sqrt{25}}{2\left(-4\right)}$$

Take the square root of 25.

$$x=\frac{-19±5}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-19±5}{-8}$$

Now solve the equation x=\frac{-19±5}{-8} when ± is plus. Add -19 to 5.

$$x=-\frac{14}{-8}$$

Reduce the fraction \frac{-14}{-8} to lowest terms by extracting and canceling out 2.

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

Now solve the equation x=\frac{-19±5}{-8} when ± is minus. Subtract 5 from -19.

$$x=-\frac{24}{-8}$$

Divide -24 by -8.

$$x=3$$

The equation is now solved.

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

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.

$$-4x^{2}+19x=21$$

Divide both sides by -4.

$$\frac{-4x^{2}+19x}{-4}=\frac{21}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{19}{-4}x=\frac{21}{-4}$$

Divide 19 by -4.

$$x^{2}-\frac{19}{4}x=\frac{21}{-4}$$

Divide 21 by -4.

$$x^{2}-\frac{19}{4}x=-\frac{21}{4}$$

Divide -\frac{19}{4}, the coefficient of the x term, by 2 to get -\frac{19}{8}. Then add the square of -\frac{19}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{19}{4}x+\left(-\frac{19}{8}\right)^{2}=-\frac{21}{4}+\left(-\frac{19}{8}\right)^{2}$$

Square -\frac{19}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{19}{4}x+\frac{361}{64}=-\frac{21}{4}+\frac{361}{64}$$

Add -\frac{21}{4} to \frac{361}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{19}{4}x+\frac{361}{64}=\frac{25}{64}$$

Factor x^{2}-\frac{19}{4}x+\frac{361}{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}.

$$\left(x-\frac{19}{8}\right)^{2}=\frac{25}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{19}{8}\right)^{2}}=\sqrt{\frac{25}{64}}$$

Simplify.

$$x-\frac{19}{8}=\frac{5}{8} x-\frac{19}{8}=-\frac{5}{8}$$

Add \frac{19}{8} to both sides of the equation.

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