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PSAT Math : How to find the square of a sum

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Example Questions

Example Question #1 : Squaring / Square Roots / Radicals

Simplify the radical.

$\sqrt{3283}$ $\displaystyle \sqrt{3283}$

Possible Answers:

$7\sqrt{67}$ $\displaystyle 7\sqrt{67}$

$7\sqrt{63}$ $\displaystyle 7\sqrt{63}$

$57.3$ $\displaystyle 57.3$

$67\sqrt{49}$ $\displaystyle 67\sqrt{49}$

$56$ $\displaystyle 56$

Correct answer:

$7\sqrt{67}$ $\displaystyle 7\sqrt{67}$

Explanation:

We can break the square root down into 2 roots of 67 and 49. 49 is a perfect square and reduces to 7.

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Example Question #61 : Exponents

Expand:

$\left ( 10x+ \frac{1}{5}\right )^{2}$ $\displaystyle \left ( 10x+ \frac{1}{5}\right )^{2}$

Possible Answers:

$100x^{2} + \frac{1}{25}$ $\displaystyle 100x^{2} + \frac{1}{25}$

$100x^{2} + \frac{1}{10}$ $\displaystyle 100x^{2} + \frac{1}{10}$

$100x^{2} + 2x + \frac{1}{25}$ $\displaystyle 100x^{2} + 2x + \frac{1}{25}$

$100x^{2} + 4x + \frac{1}{25}$ $\displaystyle 100x^{2} + 4x + \frac{1}{25}$

$100x^{2} + 2x + \frac{1}{10}$ $\displaystyle 100x^{2} + 2x + \frac{1}{10}$

Correct answer:

$100x^{2} + 4x + \frac{1}{25}$ $\displaystyle 100x^{2} + 4x + \frac{1}{25}$

Explanation:

Use the perfect square trinomial pattern, setting $A = 10x , B = \frac{1}{5}$ $\displaystyle A = 10x , B = \frac{1}{5}$:

$(A + B)^{2} = A^{2} + 2AB + B ^{2}$ $\displaystyle (A + B)^{2} = A^{2} + 2AB + B ^{2}$

$\left ( 10x + \frac{1}{5} \right )^{2} = (10x)^{2} + 2 (10x) \left ( \frac{1}{5} \right ) + \left ( \frac{1}{5} \right ) ^{2}$ $\displaystyle \left ( 10x + \frac{1}{5} \right )^{2} = (10x)^{2} + 2 (10x) \left ( \frac{1}{5} \right ) + \left ( \frac{1}{5} \right ) ^{2}$

$= 100x^{2} + 4x + \frac{1}{25}$ $\displaystyle = 100x^{2} + 4x + \frac{1}{25}$

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Example Question #3 : How To Find The Square Of A Sum

If $\left (5 x^{2 } - x + 1\right )^{2}$ $\displaystyle \left (5 x^{2 } - x + 1\right )^{2}$ is expanded, what is the coefficient of $x^{2}$ $\displaystyle x^{2}$ ?

Possible Answers:

$\displaystyle 4$

$\displaystyle 9$

$\displaystyle 11$

$\displaystyle 6$

There is no $x^{2}$ $\displaystyle x^{2}$ term in the expansion of $\left (5 x^{2 } - x + 1\right )^{2}$ $\displaystyle \left (5 x^{2 } - x + 1\right )^{2}$.

Correct answer:

$\displaystyle 11$

Explanation:

$\left (5 x^{2 } - x + 1\right )^{2}$ $\displaystyle \left (5 x^{2 } - x + 1\right )^{2}$

$= \left (5 x^{2 } - x + 1\right ) \left (5 x^{2 } - x + 1\right )$ $\displaystyle = \left (5 x^{2 } - x + 1\right ) \left (5 x^{2 } - x + 1\right )$

$= 5 x^{2 } \left (5 x^{2 } - x + 1\right ) - x \left (5 x^{2 } - x + 1\right ) + 1 \left (5 x^{2 } - x + 1\right )$ $\displaystyle = 5 x^{2 } \left (5 x^{2 } - x + 1\right ) - x \left (5 x^{2 } - x + 1\right ) + 1 \left (5 x^{2 } - x + 1\right )$

$= 5 x^{2 } \cdot 5 x^{2 } - 5 x^{2 } \cdot x + 5x^{2 } \cdot 1 - x \cdot 5 x^{2 }+ x \cdot x - x \cdot 1+ 1\cdot 5 x^{2 } - 1 \cdot x + 1\cdot 1$ $\displaystyle = 5 x^{2 } \cdot 5 x^{2 } - 5 x^{2 } \cdot x + 5x^{2 } \cdot 1 - x \cdot 5 x^{2 }+ x \cdot x - x \cdot 1+ 1\cdot 5 x^{2 } - 1 \cdot x + 1\cdot 1$

$= 25 x^{4 } - 5 x^{3 } + 5x^{2 } - 5 x^{3 }+ x ^{2 } - x + 5 x^{2 } - x + 1$ $\displaystyle = 25 x^{4 } - 5 x^{3 } + 5x^{2 } - 5 x^{3 }+ x ^{2 } - x + 5 x^{2 } - x + 1$

$= 25 x^{4 } - 5 x^{3 } - 5 x^{3 } + 5x^{2 }+ 5 x^{2 } + x ^{2 } - x - x + 1$ $\displaystyle = 25 x^{4 } - 5 x^{3 } - 5 x^{3 } + 5x^{2 }+ 5 x^{2 } + x ^{2 } - x - x + 1$

$= 25 x^{4 } -10 x^{3 } + 11 x^{2 } -2 x + 1$ $\displaystyle = 25 x^{4 } -10 x^{3 } + 11 x^{2 } -2 x + 1$

The coefficient of $x^{2}$ $\displaystyle x^{2}$ is therefore 11.

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Example Question #503 : Algebra

If $\left (5 x^{2 } - x + 1\right )^{2}$ $\displaystyle \left (5 x^{2 } - x + 1\right )^{2}$ is expanded, what is the coefficient of $x^{3}$ $\displaystyle x^{3}$ ?

Possible Answers:

$\displaystyle 10$

$\displaystyle 2$

$\displaystyle 1$

There is no $x^{3}$ $\displaystyle x^{3}$ term in the expansion of $\left (5 x^{2 } - x + 1\right )^{2}$ $\displaystyle \left (5 x^{2 } - x + 1\right )^{2}$.

$\displaystyle 5$

Correct answer:

$\displaystyle 10$

Explanation:

$\left (5 x^{2 } - x + 1\right )^{2}$ $\displaystyle \left (5 x^{2 } - x + 1\right )^{2}$

$= \left (5 x^{2 } - x + 1\right ) \left (5 x^{2 } - x + 1\right )$ $\displaystyle = \left (5 x^{2 } - x + 1\right ) \left (5 x^{2 } - x + 1\right )$

$= 25 x^{4 } - 5 x^{3 } + 5x^{2 } - 5 x^{3 }+ x ^{2 } - x + 5 x^{2 } - x + 1$ $\displaystyle = 25 x^{4 } - 5 x^{3 } + 5x^{2 } - 5 x^{3 }+ x ^{2 } - x + 5 x^{2 } - x + 1$

$= 25 x^{4 } - 5 x^{3 } - 5 x^{3 } + 5x^{2 }+ 5 x^{2 } + x ^{2 } - x - x + 1$ $\displaystyle = 25 x^{4 } - 5 x^{3 } - 5 x^{3 } + 5x^{2 }+ 5 x^{2 } + x ^{2 } - x - x + 1$

$= 25 x^{4 } -10 x^{3 } + 11 x^{2 } -2 x + 1$ $\displaystyle = 25 x^{4 } -10 x^{3 } + 11 x^{2 } -2 x + 1$

The coefficient of $x^{3}$ $\displaystyle x^{3}$ is therefore 10.

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Example Question #3 : Squaring / Square Roots / Radicals

Expand:

$\left ( 10x+ 0.8\right )^{2}$ $\displaystyle \left ( 10x+ 0.8\right )^{2}$

Possible Answers:

$100x^{2} + 1.6x + 6.4$ $\displaystyle 100x^{2} + 1.6x + 6.4$

$100x^{2}+8x + 0.64$ $\displaystyle 100x^{2}+8x + 0.64$

$100x^{2} + 16x + 0.64$ $\displaystyle 100x^{2} + 16x + 0.64$

$100x^{2} +0. 16x + 0.064$ $\displaystyle 100x^{2} +0. 16x + 0.064$

$100x^{2} +0.8x+ 6.4$ $\displaystyle 100x^{2} +0.8x+ 6.4$

Correct answer:

$100x^{2} + 16x + 0.64$ $\displaystyle 100x^{2} + 16x + 0.64$

Explanation:

Use the perfect square trinomial pattern, setting $\displaystyle A = 10x , B = 0.8$:

$(A + B)^{2} = A^{2} + 2AB + B ^{2}$ $\displaystyle (A + B)^{2} = A^{2} + 2AB + B ^{2}$

$(10x + 0.8)^{2} = \left (10x \right )^{2} + 2 \cdot 10x \cdot 0.8 + 0.8^{2}$ $\displaystyle (10x + 0.8)^{2} = \left (10x \right )^{2} + 2 \cdot 10x \cdot 0.8 + 0.8^{2}$

$(10x + 0.8)^{2} = 100x^{2} + 16x + 0.64$ $\displaystyle (10x + 0.8)^{2} = 100x^{2} + 16x + 0.64$

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PSAT Math : How to find the square of a sum

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #1 : Squaring / Square Roots / Radicals

Example Question #61 : Exponents

Example Question #3 : How To Find The Square Of A Sum

Example Question #503 : Algebra

Example Question #3 : Squaring / Square Roots / Radicals

All PSAT Math Resources

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