Solve 13u+25=3u-u^2

$$13 u + 25 = 3 u - u ^ { 2 }$$

Solve for u

$$u=-5$$

Steps Using Factoring

Subtract 3u from both sides.

$$13u+25-3u=-u^{2}$$

Combine 13u and -3u to get 10u.

$$10u+25=-u^{2}$$

Add u^{2} to both sides.

$$10u+25+u^{2}=0$$

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

$$u^{2}+10u+25=0$$

To solve the equation, factor u^{2}+10u+25 using formula u^{2}+\left(a+b\right)u+ab=\left(u+a\right)\left(u+b\right). To find a and b, set up a system to be solved.

$$a+b=10 ab=25$$

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 25.

$$1,25 5,5$$

Calculate the sum for each pair.

$$1+25=26 5+5=10$$

The solution is the pair that gives sum 10.

$$a=5 b=5$$

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

$$\left(u+5\right)\left(u+5\right)$$

Rewrite as a binomial square.

$$\left(u+5\right)^{2}$$

To find equation solution, solve u+5=0.

$$u=-5$$

Steps Using Factoring By Grouping

Subtract 3u from both sides.

$$13u+25-3u=-u^{2}$$

Combine 13u and -3u to get 10u.

$$10u+25=-u^{2}$$

Add u^{2} to both sides.

$$10u+25+u^{2}=0$$

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

$$u^{2}+10u+25=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as u^{2}+au+bu+25. To find a and b, set up a system to be solved.

$$a+b=10 ab=1\times 25=25$$

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 25.

$$1,25 5,5$$

Calculate the sum for each pair.

$$1+25=26 5+5=10$$

The solution is the pair that gives sum 10.

$$a=5 b=5$$

Rewrite u^{2}+10u+25 as \left(u^{2}+5u\right)+\left(5u+25\right).

$$\left(u^{2}+5u\right)+\left(5u+25\right)$$

Factor out u in the first and 5 in the second group.

$$u\left(u+5\right)+5\left(u+5\right)$$

Factor out common term u+5 by using distributive property.

$$\left(u+5\right)\left(u+5\right)$$

Rewrite as a binomial square.

$$\left(u+5\right)^{2}$$

To find equation solution, solve u+5=0.

$$u=-5$$

Steps Using the Quadratic Formula

Subtract 3u from both sides.

$$13u+25-3u=-u^{2}$$

Combine 13u and -3u to get 10u.

$$10u+25=-u^{2}$$

Add u^{2} to both sides.

$$10u+25+u^{2}=0$$

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.

$$u^{2}+10u+25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 10 for b, and 25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$u=\frac{-10±\sqrt{10^{2}-4\times 25}}{2}$$

Square 10.

$$u=\frac{-10±\sqrt{100-4\times 25}}{2}$$

Multiply -4 times 25.

$$u=\frac{-10±\sqrt{100-100}}{2}$$

Add 100 to -100.

$$u=\frac{-10±\sqrt{0}}{2}$$

Take the square root of 0.

$$u=-\frac{10}{2}$$

Divide -10 by 2.

$$u=-5$$

Steps for Completing the Square

Subtract 3u from both sides.

$$13u+25-3u=-u^{2}$$

Combine 13u and -3u to get 10u.

$$10u+25=-u^{2}$$

Add u^{2} to both sides.

$$10u+25+u^{2}=0$$

Subtract 25 from both sides. Anything subtracted from zero gives its negation.

$$10u+u^{2}=-25$$

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.

$$u^{2}+10u=-25$$

Divide 10, the coefficient of the x term, by 2 to get 5. Then add the square of 5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$u^{2}+10u+5^{2}=-25+5^{2}$$

Square 5.

$$u^{2}+10u+25=-25+25$$

Add -25 to 25.

$$u^{2}+10u+25=0$$

Factor u^{2}+10u+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(u+5\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(u+5\right)^{2}}=\sqrt{0}$$

Simplify.

$$u+5=0 u+5=0$$

Subtract 5 from both sides of the equation.

$$u=-5 u=-5$$

The equation is now solved. Solutions are the same.

$$u=-5$$