Solve 5u=4u^2-6

$$5 u = 4 u ^ { 2 } - 6$$

Solve for u

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

$$u=2$$

Steps Using Factoring By Grouping

Subtract 4u^{2} from both sides.

$$5u-4u^{2}=-6$$

Add 6 to both sides.

$$5u-4u^{2}+6=0$$

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

$$-4u^{2}+5u+6=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -4u^{2}+au+bu+6. To find a and b, set up a system to be solved.

$$a+b=5 ab=-4\times 6=-24$$

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 -24.

$$-1,24 -2,12 -3,8 -4,6$$

Calculate the sum for each pair.

$$-1+24=23 -2+12=10 -3+8=5 -4+6=2$$

The solution is the pair that gives sum 5.

$$a=8 b=-3$$

Rewrite -4u^{2}+5u+6 as \left(-4u^{2}+8u\right)+\left(-3u+6\right).

$$\left(-4u^{2}+8u\right)+\left(-3u+6\right)$$

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

$$4u\left(-u+2\right)+3\left(-u+2\right)$$

Factor out common term -u+2 by using distributive property.

$$\left(-u+2\right)\left(4u+3\right)$$

To find equation solutions, solve -u+2=0 and 4u+3=0.

$$u=2 u=-\frac{3}{4}$$

Steps Using the Quadratic Formula

Subtract 4u^{2} from both sides.

$$5u-4u^{2}=-6$$

Add 6 to both sides.

$$5u-4u^{2}+6=0$$

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.

$$-4u^{2}+5u+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 5 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$u=\frac{-5±\sqrt{5^{2}-4\left(-4\right)\times 6}}{2\left(-4\right)}$$

Square 5.

$$u=\frac{-5±\sqrt{25-4\left(-4\right)\times 6}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$u=\frac{-5±\sqrt{25+16\times 6}}{2\left(-4\right)}$$

Multiply 16 times 6.

$$u=\frac{-5±\sqrt{25+96}}{2\left(-4\right)}$$

Add 25 to 96.

$$u=\frac{-5±\sqrt{121}}{2\left(-4\right)}$$

Take the square root of 121.

$$u=\frac{-5±11}{2\left(-4\right)}$$

Multiply 2 times -4.

$$u=\frac{-5±11}{-8}$$

Now solve the equation u=\frac{-5±11}{-8} when ± is plus. Add -5 to 11.

$$u=\frac{6}{-8}$$

Reduce the fraction \frac{6}{-8} to lowest terms by extracting and canceling out 2.

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

Now solve the equation u=\frac{-5±11}{-8} when ± is minus. Subtract 11 from -5.

$$u=-\frac{16}{-8}$$

Divide -16 by -8.

$$u=2$$

The equation is now solved.

$$u=-\frac{3}{4} u=2$$

Steps for Completing the Square

Subtract 4u^{2} from both sides.

$$5u-4u^{2}=-6$$

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.

$$-4u^{2}+5u=-6$$

Divide both sides by -4.

$$\frac{-4u^{2}+5u}{-4}=-\frac{6}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$u^{2}+\frac{5}{-4}u=-\frac{6}{-4}$$

Divide 5 by -4.

$$u^{2}-\frac{5}{4}u=-\frac{6}{-4}$$

Reduce the fraction \frac{-6}{-4} to lowest terms by extracting and canceling out 2.

$$u^{2}-\frac{5}{4}u=\frac{3}{2}$$

Divide -\frac{5}{4}, the coefficient of the x term, by 2 to get -\frac{5}{8}. Then add the square of -\frac{5}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$u^{2}-\frac{5}{4}u+\left(-\frac{5}{8}\right)^{2}=\frac{3}{2}+\left(-\frac{5}{8}\right)^{2}$$

Square -\frac{5}{8} by squaring both the numerator and the denominator of the fraction.

$$u^{2}-\frac{5}{4}u+\frac{25}{64}=\frac{3}{2}+\frac{25}{64}$$

Add \frac{3}{2} to \frac{25}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$u^{2}-\frac{5}{4}u+\frac{25}{64}=\frac{121}{64}$$

Factor u^{2}-\frac{5}{4}u+\frac{25}{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}.

$$\left(u-\frac{5}{8}\right)^{2}=\frac{121}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(u-\frac{5}{8}\right)^{2}}=\sqrt{\frac{121}{64}}$$

Simplify.

$$u-\frac{5}{8}=\frac{11}{8} u-\frac{5}{8}=-\frac{11}{8}$$

Add \frac{5}{8} to both sides of the equation.

$$u=2 u=-\frac{3}{4}$$