Mathski — Solutions of Quadratics

Solutions of Quadratics

IB Mathematics AA · SL & HL · Shadow Worksheet

Practice

Name

Question 1

Find the value(s) of \(k\) for which the equation has equal roots:

(a)\(x^2+kx+16=0\)

(b)\(kx^2-6x+k=0\)

(c)\(3x^2+(k+1)x+3=0\)

(a)

Set \(\Delta=0\)

\(k^2-64=0\)

Answer

\(k=\pm8\)

(b)

Set \(\Delta=0\)

\(36-4k^2=0\Rightarrow k^2=9\)

Answer

\(k=\pm3\)

(c)

Set \(\Delta=0\)

\((k+1)^2-36=0\Rightarrow(k+7)(k-5)=0\)

Answer

\(k=5\) or \(k=-7\)

Question 2

The quadratic \(y=x^2-bx+4\) does not intersect the \(x\)-axis. Find the range of values of \(b\).

Condition: \(\Delta<0\)

\(b^2-16<0\Rightarrow -4

Answer

\(-4

Question 3

Solve the following quadratic inequalities:

(a)\(x^2-x-6>0\)

(b)\(2x^2+x-3\leq0\)

(a)

Roots

\((x-3)(x+2)=0\Rightarrow x=3,-2\)

Answer

\(x<-2\) or \(x>3\)

(b)

Roots

\((2x+3)(x-1)=0\Rightarrow x=- frac{3}{2}, 1\)

Answer

\(- frac{3}{2}\leq x\leq 1\)

Question 4

A rectangle has perimeter 40 cm and area 91 cm². Find the dimensions.

Setup

Let width \(=x\); length \(=20-x\). Area: \(x(20-x)=91\Rightarrow x^2-20x+91=0\)

Solve

\((x-7)(x-13)=0\)

Answer

\(7\) cm by \(13\) cm

Question 5

For what values of \(m\) does the line \(y=mx+3\) intersect the parabola \(y=x^2+1\) at two distinct points?

Set equal

\(x^2-mx-2=0\); for two distinct intersections: \(\Delta>0\Rightarrow m^2+8>0\)

Answer

All real values of \(m\) (since \(m^2+8>0\) always)

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