Solve Factorise (2x+ 1 3 )^ 2 -(x- 1 2 )^ 2 * | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$( 2 x + \frac { 1 } { 3 } ) ^ { 2 } - ( x - \frac { 1 } { 2 } ) ^ { 2 }$$

Evaluate

$3x^{2}+\frac{7x}{3}-\frac{5}{36}$

Solution Steps

Use binomial theorem $\left(a+b\right)^{2}=a^{2}+2ab+b^{2}$ to expand $\left(2x+\frac{1}{3}\right)^{2}$.

$$4x^{2}+\frac{4}{3}x+\frac{1}{9}-\left(x-\frac{1}{2}\right)^{2}$$

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

$$4x^{2}+\frac{4}{3}x+\frac{1}{9}-\left(x^{2}-x+\frac{1}{4}\right)$$

To find the opposite of $x^{2}-x+\frac{1}{4}$, find the opposite of each term.

$$4x^{2}+\frac{4}{3}x+\frac{1}{9}-x^{2}+x-\frac{1}{4}$$

Combine $4x^{2}$ and $-x^{2}$ to get $3x^{2}$.

$$3x^{2}+\frac{4}{3}x+\frac{1}{9}+x-\frac{1}{4}$$

Combine $\frac{4}{3}x$ and $x$ to get $\frac{7}{3}x$.

$$3x^{2}+\frac{7}{3}x+\frac{1}{9}-\frac{1}{4}$$

Subtract $\frac{1}{4}$ from $\frac{1}{9}$ to get $-\frac{5}{36}$.

$$3x^{2}+\frac{7}{3}x-\frac{5}{36}$$

Expand

$3x^{2}+\frac{7x}{3}-\frac{5}{36}$

Solution Steps

Use binomial theorem $\left(a+b\right)^{2}=a^{2}+2ab+b^{2}$ to expand $\left(2x+\frac{1}{3}\right)^{2}$.

$$4x^{2}+\frac{4}{3}x+\frac{1}{9}-\left(x-\frac{1}{2}\right)^{2}$$

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

$$4x^{2}+\frac{4}{3}x+\frac{1}{9}-\left(x^{2}-x+\frac{1}{4}\right)$$

To find the opposite of $x^{2}-x+\frac{1}{4}$, find the opposite of each term.

$$4x^{2}+\frac{4}{3}x+\frac{1}{9}-x^{2}+x-\frac{1}{4}$$

Combine $4x^{2}$ and $-x^{2}$ to get $3x^{2}$.

$$3x^{2}+\frac{4}{3}x+\frac{1}{9}+x-\frac{1}{4}$$

Combine $\frac{4}{3}x$ and $x$ to get $\frac{7}{3}x$.

$$3x^{2}+\frac{7}{3}x+\frac{1}{9}-\frac{1}{4}$$

Subtract $\frac{1}{4}$ from $\frac{1}{9}$ to get $-\frac{5}{36}$.

$$3x^{2}+\frac{7}{3}x-\frac{5}{36}$$