Practice Questions
Test your command of algebraic identities and equations with these common exam questions.
If $a + b = 10$ and $ab = 21$, find the value of $a^2 + b^2$.
Using identity $(a+b)^2 = a^2+b^2+2ab$. $10^2 = a^2+b^2+2(21) \Rightarrow 100 = a^2+b^2+42 \Rightarrow a^2+b^2 = 58$.
Solve for $x$: $x^2 - 7x + 12 = 0$.
Split the middle term: $x^2-3x-4x+12=0 \Rightarrow x(x-3)-4(x-3)=0 \Rightarrow (x-4)(x-3)=0 \Rightarrow x=3, 4$.
If $x + 1/x = 4$, find $x^3 + 1/x^3$.
Using $k^3-3k$: $4^3 - 3(4) = 64 - 12 = 52$.
What are the roots of the equation $x^2 + x - 6 = 0$?
$x^2+3x-2x-6=0 \Rightarrow x(x+3)-2(x+3)=0 \Rightarrow (x-2)(x+3)=0 \Rightarrow x=2, -3$.
Simplify $\frac{x^2 - 16}{x - 4}$.
$x^2-16 = (x-4)(x+4)$. So $(x-4)(x+4)/(x-4) = x+4$.
Solve $3x - 7 = 2x + 5$.
$3x-2x = 5+7 \Rightarrow x = 12$.
Factorise $x^3 - 27$.
Using $a^3-b^3 = (a-b)(a^2+ab+b^2)$ with $a=x$, $b=3$: $x^3-27 = (x-3)(x^2+3x+9)$.
If $x - 1/x = 3$, find $x^2 + 1/x^2$.
Squaring: $(x-1/x)^2 = x^2+1/x^2-2 \Rightarrow 9 = x^2+1/x^2 -2 \Rightarrow x^2+1/x^2 = 11$.
If $x + 1/x = 5$, find $x^5 + 1/x^5$.
$x^2+1/x^2=5^2-2=23$, $x^3+1/x^3=5^3-3(5)=110$. Multiplying: $(x^3+1/x^3)(x^2+1/x^2) = x^5+1/x^5+x+1/x \Rightarrow 110\times23 = x^5+1/x^5+5 \Rightarrow x^5+1/x^5 = 2530-5 = 2525$.
Solve the inequality $x^2 - 4x + 3 > 0$.
$(x-1)(x-3) > 0$. Product positive when both factors positive ($x>3$) or both negative ($x<1$). So $x<1$ or $x>3$.
Premium Content
Unlock Algebra Quiz and all premium lessons with a subscription.
From ₹199.99/year — See plans