Algebra Examples

${(2 x - 1)}^{2} = (x + 1)$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Simplify the left side.

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Step 1.1.1

Remove parentheses.

${(2 x - 1)}^{2} = x + 1$

Step 1.2

Move all the expressions to the left side of the equation.

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Step 1.2.1

Subtract $x$ from both sides of the equation.

${(2 x - 1)}^{2} - x = 1$

Step 1.2.2

Subtract $1$ from both sides of the equation.

${(2 x - 1)}^{2} - x - 1 = 0$

Step 1.3

Simplify ${(2 x - 1)}^{2} - x - 1$ .

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Rewrite ${(2 x - 1)}^{2}$ as $(2 x - 1) (2 x - 1)$ .

$(2 x - 1) (2 x - 1) - x - 1 = 0$

Step 1.3.1.2

Expand $(2 x - 1) (2 x - 1)$ using the FOIL Method.

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Step 1.3.1.2.1

Apply the distributive property.

$2 x (2 x - 1) - 1 (2 x - 1) - x - 1 = 0$

Step 1.3.1.2.2

Apply the distributive property.

$2 x (2 x) + 2 x \cdot - 1 - 1 (2 x - 1) - x - 1 = 0$

Step 1.3.1.2.3

Apply the distributive property.

$2 x (2 x) + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1 - x - 1 = 0$

Step 1.3.1.3

Simplify and combine like terms.

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Step 1.3.1.3.1

Simplify each term.

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Step 1.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$2 \cdot 2 x \cdot x + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1 - x - 1 = 0$

Step 1.3.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.3.1.3.1.2.1

Move $x$ .

$2 \cdot 2 (x \cdot x) + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1 - x - 1 = 0$

Step 1.3.1.3.1.2.2

Multiply $x$ by $x$ .

$2 \cdot 2 x^{2} + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1 - x - 1 = 0$

Step 1.3.1.3.1.3

Multiply $2$ by $2$ .

$4 x^{2} + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1 - x - 1 = 0$

Step 1.3.1.3.1.4

Multiply $- 1$ by $2$ .

$4 x^{2} - 2 x - 1 (2 x) - 1 \cdot - 1 - x - 1 = 0$

Step 1.3.1.3.1.5

Multiply $2$ by $- 1$ .

$4 x^{2} - 2 x - 2 x - 1 \cdot - 1 - x - 1 = 0$

Step 1.3.1.3.1.6

Multiply $- 1$ by $- 1$ .

$4 x^{2} - 2 x - 2 x + 1 - x - 1 = 0$

Step 1.3.1.3.2

Subtract $2 x$ from $- 2 x$ .

$4 x^{2} - 4 x + 1 - x - 1 = 0$

Step 1.3.2

Combine the opposite terms in $4 x^{2} - 4 x + 1 - x - 1$ .

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Step 1.3.2.1

Subtract $1$ from $1$ .

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

Step 1.3.2.2

Add $4 x^{2} - 4 x - x$ and $0$ .

$4 x^{2} - 4 x - x = 0$

Step 1.3.3

Subtract $x$ from $- 4 x$ .

$4 x^{2} - 5 x = 0$

Step 2

Once the quadratic is in standard form, the values of $a$ , $b$ , and $c$ can be found.

$a x^{2} + b x + c$

Step 3

Use the standard form of the equation to find $a$ , $b$ , and $c$ for this quadratic.

$a = 4$ , $b = - 5$ , $c = 0$