Solve 49x-45=left(x+1right)^2

$$49x-45= { \left(x+1 \right) }^{ 2 }$$

Solve for x

$$x=1$$

$$x=46$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$49x-45=x^{2}+2x+1$$

Subtract x^{2} from both sides.

$$49x-45-x^{2}=2x+1$$

Subtract 2x from both sides.

$$49x-45-x^{2}-2x=1$$

Combine 49x and -2x to get 47x.

$$47x-45-x^{2}=1$$

Subtract 1 from both sides.

$$47x-45-x^{2}-1=0$$

Subtract 1 from -45 to get -46.

$$47x-46-x^{2}=0$$

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

$$-x^{2}+47x-46=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-46. To find a and b, set up a system to be solved.

$$a+b=47 ab=-\left(-46\right)=46$$

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 46.

$$1,46 2,23$$

Calculate the sum for each pair.

$$1+46=47 2+23=25$$

The solution is the pair that gives sum 47.

$$a=46 b=1$$

Rewrite -x^{2}+47x-46 as \left(-x^{2}+46x\right)+\left(x-46\right).

$$\left(-x^{2}+46x\right)+\left(x-46\right)$$

Factor out -x in -x^{2}+46x.

$$-x\left(x-46\right)+x-46$$

Factor out common term x-46 by using distributive property.

$$\left(x-46\right)\left(-x+1\right)$$

To find equation solutions, solve x-46=0 and -x+1=0.

$$x=46 x=1$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$49x-45=x^{2}+2x+1$$

Subtract x^{2} from both sides.

$$49x-45-x^{2}=2x+1$$

Subtract 2x from both sides.

$$49x-45-x^{2}-2x=1$$

Combine 49x and -2x to get 47x.

$$47x-45-x^{2}=1$$

Subtract 1 from both sides.

$$47x-45-x^{2}-1=0$$

Subtract 1 from -45 to get -46.

$$47x-46-x^{2}=0$$

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.

$$-x^{2}+47x-46=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 47 for b, and -46 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-47±\sqrt{47^{2}-4\left(-1\right)\left(-46\right)}}{2\left(-1\right)}$$

Square 47.

$$x=\frac{-47±\sqrt{2209-4\left(-1\right)\left(-46\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-47±\sqrt{2209+4\left(-46\right)}}{2\left(-1\right)}$$

Multiply 4 times -46.

$$x=\frac{-47±\sqrt{2209-184}}{2\left(-1\right)}$$

Add 2209 to -184.

$$x=\frac{-47±\sqrt{2025}}{2\left(-1\right)}$$

Take the square root of 2025.

$$x=\frac{-47±45}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-47±45}{-2}$$

Now solve the equation x=\frac{-47±45}{-2} when ± is plus. Add -47 to 45.

$$x=-\frac{2}{-2}$$

Divide -2 by -2.

$$x=1$$

Now solve the equation x=\frac{-47±45}{-2} when ± is minus. Subtract 45 from -47.

$$x=-\frac{92}{-2}$$

Divide -92 by -2.

$$x=46$$

The equation is now solved.

$$x=1 x=46$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$49x-45=x^{2}+2x+1$$

Subtract x^{2} from both sides.

$$49x-45-x^{2}=2x+1$$

Subtract 2x from both sides.

$$49x-45-x^{2}-2x=1$$

Combine 49x and -2x to get 47x.

$$47x-45-x^{2}=1$$

Add 45 to both sides.

$$47x-x^{2}=1+45$$

Add 1 and 45 to get 46.

$$47x-x^{2}=46$$

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.

$$-x^{2}+47x=46$$

Divide both sides by -1.

$$\frac{-x^{2}+47x}{-1}=\frac{46}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{47}{-1}x=\frac{46}{-1}$$

Divide 47 by -1.

$$x^{2}-47x=\frac{46}{-1}$$

Divide 46 by -1.

$$x^{2}-47x=-46$$

Divide -47, the coefficient of the x term, by 2 to get -\frac{47}{2}. Then add the square of -\frac{47}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-47x+\left(-\frac{47}{2}\right)^{2}=-46+\left(-\frac{47}{2}\right)^{2}$$

Square -\frac{47}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-47x+\frac{2209}{4}=-46+\frac{2209}{4}$$

Add -46 to \frac{2209}{4}.

$$x^{2}-47x+\frac{2209}{4}=\frac{2025}{4}$$

Factor x^{2}-47x+\frac{2209}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{47}{2}\right)^{2}=\frac{2025}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{47}{2}\right)^{2}}=\sqrt{\frac{2025}{4}}$$

Simplify.

$$x-\frac{47}{2}=\frac{45}{2} x-\frac{47}{2}=-\frac{45}{2}$$

Add \frac{47}{2} to both sides of the equation.

$$x=46 x=1$$