SAT Algebra Perfect Score

To get a perfect score on SAT Math, you need to:

• Get every single problem correct.
• Have complete mastery of all of the SAT skills
• Remember the Tips and use them
• Figure out your common mistakes and avoid them

If \(P^2=\sqrt{76+10\sqrt{3}}+5\sqrt{52-14\sqrt{3}},\) what is the negative number \(P?\)

(A) \(\ \ -4\)
(B) \(\ \ -5\)
(C) \(\ \ -5.5\)
(D) \(\ \ -6\)
(E) \(\ \ -7\)

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Correct Answer: D

Solution:

Using the technique of completing the square, we can rewrite the given equation as follows: \[\begin{align} P^2 &=\sqrt{76+10\sqrt{3}}+5\sqrt{52-14\sqrt{3}}\\ &=\sqrt{(1+5\sqrt{3})^2}+5\sqrt{(7-\sqrt{3})^2}\\ &=(1+5\sqrt{3})+5(7-\sqrt{3})\\ &=36\\ \Rightarrow P&=\pm6. \end{align}\] Since \(P\) is negative, \(P=-6.\)

Therefore, the correct answer is (D).

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Incorrect Choices:

(A), (B), (C) and (E)
The solution explains why these choices are wrong.

\(a, b\) and \(c\) are real numbers such that \[\begin{array}&a+b+c=2, &2^a+2^b+2^c=\frac{19}{2}, &2^{-a}+2^{-b}+2^{-c}=\frac{25}{8}.\end{array}\] What is the value of \(4^a+4^b+4^c?\)

(A) \( \ \ \frac{257}{4} \)
(B) \( \ \ \frac{261}{4} \)
(C) \( \ \ \frac{265}{4} \)
(D) \(\ \ \frac{269}{4} \)
(E) \( \ \ \frac{273}{4} \)

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Correct Answer: B

Solution:

We rewrite \(4^a+4^b+4^c\) as follows to obtain: \[\begin{align} 4^a+4^b+4^c &=\left(2^a\right)^2+\left(2^b\right)^2+\left(2^c\right)^2\\ &=\left(2^a+2^b+2^c\right)^2-2\left(2^a2^b+2^b2^c+2^c2^a\right)\\ &=\left(2^a+2^b+2^c\right)^2-\left(2^{a+b+1}+2^{b+c+1}+2^{c+a+1}\right)\\ &=\left(2^a+2^b+2^c\right)^2-\left(2^{3-c}+2^{3-a}+2^{3-b}\right)\\ &=\left(2^a+2^b+2^c\right)^2-2^3\cdot \left(2^{-c}+2^{-a}+2^{-b}\right)\\ &=\left(\frac{19}{2}\right)^2-8\cdot \frac{25}{8}=\frac{361-100}{4}=\frac{261}{4}. \end{align}\]

Therefore, the correct answer is (B).

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Incorrect Choices:

(A), (C), (D) and (E)
The solution explains why these choices are wrong.

\(x, y\) and \(z\) are real numbers such that \[\begin{array} &2x+y+z=10, &2x^2-yz=-14.\end{array}\] If \(m\) and \(M\) are the minimum and maximum values of \(xy+yz+zx,\) respectively, what is \(M-m?\)

(A) \( \ \ 100\)
(B) \( \ \ 110\)
(C) \( \ \ 120\)
(D) \(\ \ 130\)
(E) \( \ \ 140\)

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Correct Answer: C

Solution:

Observe that \(y+z=-2x+10=-2(x-5)\) and \(yz=2x^2+14.\) Then \(y\) and \(z\) are the two real roots of the following quadratic equation in \(t:\) \[t^2+2(x-5)+(2x^2+14).\] Since this equation has real roots, it must be true that its discriminant is non-negative: \[\begin{align} \frac{D}{4}=(x-5)^2-(2x^2+14)=-x^2-10x+11 &\ge 0 \\ \Rightarrow x^2+10x-11&\le 0 \\ (x+11)(x-1)&\le 0\\ -11&\le x \le 1. \qquad (1) \end{align}\] Now, \[\begin{align} xy+yz+zx &=yz+x(y+z)\\ &=(2x^2+14)+x\left(-2(x-5)\right)\\ &=10x+14. \qquad (2) \end{align}\] Since \(-11\le x \le 1\) from \((1),\) the range of \((2)\) is \(-96\le 10x+14 \le 24.\) Hence \(m=-96\) and \(M=24,\) which implies \(M-m=24-(-96)=120.\)

Therefore, the correct answer is (C).

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Incorrect Choices:

(A), (B), (D) and (E)
The solution explains why these choices are wrong.

Algebraic Manipulations

• Follow order of operations.

Polynomials

• \(a^{2}-b^{2}=(a-b)(a+b)\)
• \((a \pm b)^{2} = a^{2} \pm 2ab + b^{2}\)

Exponents

• Know the Rules of Exponents.
• Recognize first few perfect squares (1, 4, 9, ... 400) and cubes (1, 8, 27,... 1000).
• The square of a number is always positive.
• \(\sqrt{x^{2}} = \begin{cases} -x &\mbox{if } x < 0 \\ x & \mbox{if } x \geq 0. \\ \end{cases}\)

Change the Subject

• Know the Rules of Exponents.

Inequalities

• \(x^{2} \geq 0.\)
• Know the Properties of Inequality.
• Multiplying (or dividing) both sides of an inequality by a negative number reverses its sign.
• Know the properties of numbers between \(0\) and \(1\).

Absolute Value

• \(|x| = \begin{cases} -x &\mbox{if } x < 0 \\ x & \mbox{if } x \geq 0. \\ \end{cases}\)
• \(x^{2} \geq 0.\)
• Know the Properties of Inequality.
• Multiplying (or dividing) both sides of an inequality by a negative number reverses its sign.

SAT General Tips