Practice Questions
Test your algebraic proficiency with these quadratic equation questions.
Find the roots of the equation x² - 5x + 6 = 0.
Factoring gives $(x-2)(x-3) = 0$. So, $x=2$ or $x=3$.
What is the sum of roots for the equation 2x² + 8x - 10 = 0?
Sum of roots = $-b/a = -8/2 = -4$.
Find the discriminant for the equation x² + 4x + 4 = 0 and determine the nature of its roots.
$D = b^2 - 4ac = 4^2 - 4(1)(4) = 16 - 16 = 0$. Roots are real and equal.
If the product of roots of kx² - 4x + 12 = 0 is 3, find the value of k.
Product of roots = $c/a = 12/k$. Given $12/k = 3 \Rightarrow k = 4$.
If the roots of $x^2 - px + 12 = 0$ are in the ratio 3:4, find the value of p.
Let roots be 3k and 4k. Product = $12k^2 = 12 \Rightarrow k^2 = 1 \Rightarrow k = 1$. Roots are 3 and 4. Sum = $p = 3 + 4 = 7$.
For what value of k does the equation $x^2 + kx + 9 = 0$ have equal roots?
$Δ = k^2 - 36 = 0 \Rightarrow k = ±6$.
If one root of $x^2 - 7x + 10 = 0$ is 2, and the other root is increased by 2, find the new quadratic equation.
Original roots of $x^2 - 7x + 10 = 0$ are 2 and 5. Increasing 5 by 2 gives 7. New roots: 2 and 7. Sum = 9, product = 14. New equation: $x^2 - 9x + 14 = 0$.
The product of two consecutive positive integers is 132. Form and solve the quadratic equation.
Let integers be n and n+1. $n(n+1) = 132 \Rightarrow n^2 + n - 132 = 0 \Rightarrow (n+12)(n-11) = 0 \Rightarrow n = 11$ (positive). Integers are 11 and 12.
If the roots of $2x^2 - 5x + 3 = 0$ are α and β, find the value of $α^2 + β^2$.
$α+β = 5/2$, $αβ = 3/2$. $α^2+β^2 = (α+β)^2 - 2αβ = (5/2)^2 - 2(3/2) = 25/4 - 3 = 13/4$.
Find the condition on a such that $ax^2 - 4x + 1 = 0$ has real and distinct roots.
$Δ = 16 - 4a > 0 \Rightarrow 4a < 16 \Rightarrow a < 4$. Also $a ≠ 0$ for quadratic. So $a < 4$ and $a ≠ 0$.
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