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ISEE Upper Level Math : How to find the solution to an equation

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Example Questions

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Example Question #41 : Algebraic Concepts

Solve the equation:

$\frac{2}{3}-\frac{3}{x}=\frac{8}{x}$

Possible Answers:

6.5

4.5

5.5

Correct answer:

5.5

Explanation:

First multiply both sides by :

$\frac{2}{3}-\frac{3}{x}=\frac{8}{x}\Rightarrow 3x(\frac{2}{3})-3x(\frac{3}{x})=3x(\frac{8}{x})\Rightarrow 6x-9=24\Rightarrow 6x=33$

$\Rightarrow x=5.5$

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Example Question #42 : How To Find The Solution To An Equation

For what values of is $2\left |x+5 \right |-5< 7$ ?

Possible Answers:

-11<x<4

-11<x<2

-10<x<1

-11<x<1

-11<x<10

Correct answer:

-11<x<1

Explanation:

$\left |x \right |< a\Rightarrow -a< x< a$

We have $\left |x+5 \right |< 6$ , so:

$-6<x+5<6\Rightarrow -6-5<x+5-5<6-5\Rightarrow -11<x<1$

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Example Question #41 : How To Find The Solution To An Equation

Find :

$\left |2x+4 \right |-7>9$

Possible Answers:

$x>-6 \ or \ x<10$

-10<x<6

$x>6\: or \: x<6$

$x>6 \: or \: x<-10$

$x>6 \ or \ x<10$

Correct answer:

$x>6 \: or \: x<-10$

Explanation:

$\left |x \right |>a$ means that x<-a or x>a .

Then we can write:

$\left |2x+4 \right |>16\Rightarrow 2x+4>16$

2x+4<-16

$2x+4>16\Rightarrow 2x+4-4>16-4\Rightarrow 2x>12\Rightarrow x>6$

$2x+4<-16\Rightarrow 2x+4-4<-16-4\Rightarrow 2x<-20\Rightarrow x<-10$

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Example Question #41 : Equations

Solve the equation for :

$7\sqrt{x}+11=5$

Possible Answers:

No solution

$\frac{6}{7}$

Correct answer:

No solution

Explanation:

$7\sqrt{x}+11=5\Rightarrow 7\sqrt{x}+11-11=5-11\Rightarrow 7\sqrt{x}=-6\Rightarrow \sqrt{x}=-\frac{6}{7}$

Since $\sqrt{x}$ cannot be negative, no real number satisfies the equation.

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Example Question #741 : Isee Upper Level (Grades 9 12) Mathematics Achievement

What are the roots of the equation?

3x^2-6x-45=0

Possible Answers:

$x=3 \ or\ x=-3$

$x=-5\ or \ x=3$

$x=5 \ or\ x=-3$

$x=5 \ or \ x=3$

$x=-5 \ or \ x=-3$

Correct answer:

$x=5 \ or\ x=-3$

Explanation:

First simplify the equation by dividing both sides by :

$3x^2-6x-45=0\Rightarrow x^2-2x-15=0$

If a, b, c are real numbers with $a\neq 0$ , and if ax^2+bx+c=0 , then

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ .

The expression b^2-4ac is called the discriminant of the quadratic equation, and we say D=b^2-4ac .

We can write $x=\frac{-b\pm \sqrt{D}}{2a}$ .

In this problem we have $a=1,\ b=-2, \ and\ c=-15.$

$D=b^2-4ac=(-2)^2-4\times 1\times (-15)=4+60=64$

$x=\frac{-b\pm \sqrt{D}}{2a}=\frac{-(-2)\pm \sqrt{64}}{2\times 1}=\frac{2\pm 8}{2}$

Therefore the roots are

$x=\frac{2+8}{2}=5$ or $x=\frac{2-8}{2}=-3$ .

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Example Question #46 : How To Find The Solution To An Equation

What are the roots of the equation?

x^2=28-3x

Possible Answers:

$x=-4\ or\ x=7$

$x=4\ or \ x=-7$

$x=-4\ or\ x=-7$

x=4

$x=4 \ or\ x=7$

Correct answer:

$x=4\ or \ x=-7$

Explanation:

First rewrite the equation in the form of ax^2+bx+c=0 :

x^2+3x-28=0

If a, b, c are real numbers with $a\neq 0$ , and if ax^2+bx+c=0 , then we have:

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

The expression b^2-4ac is called the discriminant of the quadratic equation, and we say D=b^2-4ac . We can write

$x=\frac{-b\pm \sqrt{D}}{2a}$ .

In this problem we have $a=1, \ b=3,\ and\ c=-28.$

$D=b^2-4ac=(3)^2-4\times 1\times (-28)=9+112=121$

$x=\frac{-b\pm \sqrt{D}}{2a}=\frac{-3\pm \sqrt{121}}{2\times 1}=\frac{-3\pm 11}{2}$

Then the roots are $x=\frac{-3+11}{2}=4$ and $x=\frac{-3-11}{2}=-7$ .

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Example Question #47 : How To Find The Solution To An Equation

Find the roots of the equation:

3x^2+6x-11=0

Possible Answers:

$x=1+\sqrt{21}$ or $x=1-\sqrt{21}$

$x=3+\sqrt{21}$ or $x=3-\sqrt{21}$

$x=-\sqrt{42}$ or $x=\sqrt{42}$

$x=1+\sqrt{42}$ or $x=1-\sqrt{42}$

$x=1+\frac{\sqrt{42}}{3}$ or $x=1-\frac{\sqrt{42}}{3}$

Correct answer:

$x=1+\frac{\sqrt{42}}{3}$ or $x=1-\frac{\sqrt{42}}{3}$

Explanation:

If a, b, c are real numbers with $a\neq 0$ , and if ax^2+bx+c=0 , then we have

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ .

The expression b^2-4ac is called the discriminant of the quadratic equation, and we say D=b^2-4ac . We can write

$x=\frac{-b\pm \sqrt{D}}{2a}$ .

In this problem we have $a=3,\ b=6,\ and \ c=-11.$

$D=b^2-4ac=(6)^2-4\times 3\times (-11)=36+132=168$

$x=\frac{-b\pm \sqrt{D}}{2a}=\frac{-6\pm \sqrt{168}}{2\times 3}=\frac{-6\pm 2\sqrt{42}}{6}=\frac{-3\pm \sqrt{42}}{3}$

Then the roots are

$x=-\frac{3+\sqrt{42}}{3}=-1-\frac{\sqrt{42}}{3}$ and $x=\frac{3-\sqrt{42}}{3}=1-\frac{\sqrt{42}}{3}$ .

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Example Question #48 : How To Find The Solution To An Equation

Find the roots of 4x^2-64=0 .

Possible Answers:

x=-4

x=-3 or

x=-4

x=-3

x=4

x=4 or x=-4

Correct answer:

x=4 or x=-4

Explanation:

$4x^2-64=0\Rightarrow 4x^2=64\Rightarrow x^2=16\Rightarrow x=\pm \sqrt{16}=\pm 4$

Therefore, x=4 or x=-4 .

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Example Question #49 : How To Find The Solution To An Equation

Solve for :

4x^2-3x=0

Possible Answers:

x=0

$x=\frac{3}{4}$

x=0 or $x=-\frac{3}{4}$

x=1 or $x=\frac{3}{4}$

x=0 or $x=\frac{3}{4}$

Correct answer:

x=0 or $x=\frac{3}{4}$

Explanation:

We can factor out an :

$4x^2-3x=0\Rightarrow x(4x-3)=0$

Therefore, x=0 or:

$4x-3=0\Rightarrow 4x-3+3=0+3\Rightarrow 4x=3\Rightarrow x=\frac{3}{4}$

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Example Question #751 : Isee Upper Level (Grades 9 12) Mathematics Achievement

Solve the equation for :

x^2-x-20=0

Possible Answers:

$x=-5\ or\ x=4$

$x=5 \ or\ x=4$

$x=5 \ or \ x=-4$

$x=-5\ or\ x=-4$

$x=4 \ or \ x=-4$

Correct answer:

$x=5 \ or \ x=-4$

Explanation:

Sometimes you can easily factor the experssion ax^2+bx+c=0 . Then the equation can be solved by setting each factor equal to . In this problem we have:

$x^2-x-20=0\Rightarrow (x-5)(x+4)=0\Rightarrow x-5=0$ or x+4=0 .

$x-5=0\Rightarrow x=5$

$x+4=0 \Rightarrow x=-4$

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ISEE Upper Level Math : How to find the solution to an equation

Study concepts, example questions & explanations for ISEE Upper Level Math

All ISEE Upper Level Math Resources

Example Questions

Example Question #41 : Algebraic Concepts

Example Question #42 : How To Find The Solution To An Equation

Example Question #41 : How To Find The Solution To An Equation

Example Question #41 : Equations

Example Question #741 : Isee Upper Level (Grades 9 12) Mathematics Achievement

Example Question #46 : How To Find The Solution To An Equation

Example Question #47 : How To Find The Solution To An Equation

Example Question #48 : How To Find The Solution To An Equation

Example Question #49 : How To Find The Solution To An Equation

Example Question #751 : Isee Upper Level (Grades 9 12) Mathematics Achievement

All ISEE Upper Level Math Resources

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