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Algebra: Quadratic Equations Test-3

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Question 1
Which of the following is a quadratic equation?
A
$ \displaystyle {{x}^{\frac{1}{2}}}+2x+3=0$
B
$ \displaystyle \left( x-1 \right)\,\left( x+4 \right)={{x}^{2}}+1$
C
$ \displaystyle {{x}^{4}}-3x+5=0$
D
$ \displaystyle \left( 2x+1 \right)\,\left( 3x-4 \right)\,=2{{x}^{2}}+3$
Question 1 Explanation: 
By observation the highest power of x is 2 only in option (d)
Question 2
Which one of the following is a factor of
$ \displaystyle {{x}^{3}}-19x+30$
A
$ \displaystyle x-2$
B
$ \displaystyle x+2$
C
$ \displaystyle x-1$
D
$ \displaystyle x+1$
Question 2 Explanation: 
Capture
Thus, x-2 is one of the factors.
Question 3
The solution of the equation
$ \displaystyle \sqrt{25-{{x}^{2}}}=x-1$ are
A
$ \displaystyle x=3\,\,\,and\,\,x=4$
B
$ \displaystyle x=5\,\,\,\,and\,\,\,\,x=1$
C
$ \displaystyle x=-3\,\,and\,\,x=4$
D
$ \displaystyle x=4\,\,\,\,and\,\,\,x\ne -3$
Question 3 Explanation: 
For x =4 , the equation
$latex \sqrt{25-{{x}^{2}}}=x-1$ is satisfied ,
Thus options which can be correct are a, c and d. Now when we put x=-3 we find that both sides are not equal so we can conclude that (d) is the answer)
Question 4
Of the following quadratic equations
which is the one whose roots are 2 and −15?
A
$ \displaystyle {{x}^{2}}-2x+15=0$
B
$ \displaystyle {{x}^{2}}+15x-2=0$
C
$ \displaystyle {{x}^{2}}+13x-30=0$
D
$ \displaystyle {{x}^{2}}-30=0$
Question 4 Explanation: 
The factors are x-2 and x+15.
Thus the quadratic equation is
$ \begin{array}{l}(x-2)(x+15)\\={{x}^{2}}+13x-30\end{array}$
Question 5
If $ \displaystyle \begin{array}{l}{{\left( a-3 \right)}^{2}}+{{\left( b-4 \right)}^{2}}+{{\left( c-9 \right)}^{2}}=0,\,\,\\then\,\,the\,\,value\,\,of\,\,\sqrt{a+b+c}\end{array}$ is
A
-4
B
+4
C
±4
D
±2
Question 5 Explanation: 
Since the sum of three squares can only be equal to 0
if the terms are individually equal to 0
we can conclude that a=3,b=4,c=9.
The sum of a, b, c = 16.
Thus, $ \sqrt{a+b+c}=\sqrt{16}=\pm 4$
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