Solve 210=2x^2+(x*4)8

$$210 = 2 { x }^{ 2 } +(x \times 4)8$$

Solve for x

$$x=-21$$

$$x=5$$

Steps Using Factoring By Grouping

Multiply 4 and 8 to get 32.

$$210=2x^{2}+x\times 32$$

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

$$2x^{2}+x\times 32=210$$

Subtract 210 from both sides.

$$2x^{2}+x\times 32-210=0$$

Divide both sides by 2.

$$x^{2}+16x-105=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-105. To find a and b, set up a system to be solved.

$$a+b=16 ab=1\left(-105\right)=-105$$

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 -105.

$$-1,105 -3,35 -5,21 -7,15$$

Calculate the sum for each pair.

$$-1+105=104 -3+35=32 -5+21=16 -7+15=8$$

The solution is the pair that gives sum 16.

$$a=-5 b=21$$

Rewrite x^{2}+16x-105 as \left(x^{2}-5x\right)+\left(21x-105\right).

$$\left(x^{2}-5x\right)+\left(21x-105\right)$$

Factor out x in the first and 21 in the second group.

$$x\left(x-5\right)+21\left(x-5\right)$$

Factor out common term x-5 by using distributive property.

$$\left(x-5\right)\left(x+21\right)$$

To find equation solutions, solve x-5=0 and x+21=0.

$$x=5 x=-21$$

Steps Using the Quadratic Formula

Multiply 4 and 8 to get 32.

$$210=2x^{2}+x\times 32$$

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

$$2x^{2}+x\times 32=210$$

Subtract 210 from both sides.

$$2x^{2}+x\times 32-210=0$$

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.

$$2x^{2}+32x-210=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 32 for b, and -210 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-32±\sqrt{32^{2}-4\times 2\left(-210\right)}}{2\times 2}$$

Square 32.

$$x=\frac{-32±\sqrt{1024-4\times 2\left(-210\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-32±\sqrt{1024-8\left(-210\right)}}{2\times 2}$$

Multiply -8 times -210.

$$x=\frac{-32±\sqrt{1024+1680}}{2\times 2}$$

Add 1024 to 1680.

$$x=\frac{-32±\sqrt{2704}}{2\times 2}$$

Take the square root of 2704.

$$x=\frac{-32±52}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-32±52}{4}$$

Now solve the equation x=\frac{-32±52}{4} when ± is plus. Add -32 to 52.

$$x=\frac{20}{4}$$

Divide 20 by 4.

$$x=5$$

Now solve the equation x=\frac{-32±52}{4} when ± is minus. Subtract 52 from -32.

$$x=-\frac{84}{4}$$

Divide -84 by 4.

$$x=-21$$

The equation is now solved.

$$x=5 x=-21$$

Steps for Completing the Square

Multiply 4 and 8 to get 32.

$$210=2x^{2}+x\times 32$$

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

$$2x^{2}+x\times 32=210$$

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.

$$2x^{2}+32x=210$$

Divide both sides by 2.

$$\frac{2x^{2}+32x}{2}=\frac{210}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{32}{2}x=\frac{210}{2}$$

Divide 32 by 2.

$$x^{2}+16x=\frac{210}{2}$$

Divide 210 by 2.

$$x^{2}+16x=105$$

Divide 16, the coefficient of the x term, by 2 to get 8. Then add the square of 8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+16x+8^{2}=105+8^{2}$$

Square 8.

$$x^{2}+16x+64=105+64$$

Add 105 to 64.

$$x^{2}+16x+64=169$$

Factor x^{2}+16x+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}.

$$\left(x+8\right)^{2}=169$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+8\right)^{2}}=\sqrt{169}$$

Simplify.

$$x+8=13 x+8=-13$$

Subtract 8 from both sides of the equation.

$$x=5 x=-21$$