Chapter 10

Solve the equations (a) \(x^{2}+2 x-8=0\) (b) \(3 x^{2}-11 x-4=0\) by factorization.

Determine the roots of (a) \(x^{2}-6 x+9=0\), and (b) \(4 x^{2}-25=0\), by factorization.

Solve the following quadratic equations by factorizing: (a) \(4 x^{2}+8 x+3=0\) (b) \(15 x^{2}+2 x-8=0\).

The roots of a quadratic equation are \(\frac{1}{3}\) and \(-2\). Determine the equation.

Find the equations in \(x\) whose roots are (a) 5 and \(-5\) (b) \(1.2\) and \(-0.4\).

Solve \(2 x^{2}+5 x=3\) by 'completing the square'.

Solve \(2 x^{2}+9 x+8=0\), correct to 3 significant figures, by 'completing the square'.

By 'completing the square', solve the quadratic equation \(4.6 y^{2}+3.5 y-1.75=0\), correct to 3 decimal places.

Solve (a) \(x^{2}+2 x-8=0\) and (b) \(3 x^{2}-11 x-4=0\) by using the quadratic formula.

Solve \(4 x^{2}+7 x+2=0\) giving the roots correct to 2 decimal places.

Use the quadratic formula to solve \(\frac{x+2}{4}+\frac{3}{x-1}=7\) correct to 4 significant figures.

The area of a rectangle is \(23.6 \mathrm{~cm}^{2}\) and its width is \(3.10 \mathrm{~cm}\) shorter than its length. Determine the dimensions of the rectangle, correct to 3 significant figures.

Calculate the diameter of a solid cylinder which has a height of \(82.0 \mathrm{~cm}\) and a total surface area of \(2.0 \mathrm{~m}^{2}\).

The height \(s\) metres of a mass projected vertically upwards at time \(t\) seconds is \(s=u t-\frac{1}{2} g t^{2}\). Determine how long the mass will take after being projected to reach a height of \(16 \mathrm{~m}\) (a) on the ascent and (b) on the descent, when \(u=30 \mathrm{~m} / \mathrm{s}\) and \(g=9.81 \mathrm{~m} / \mathrm{s}^{2}\).

A shed is \(4.0 \mathrm{~m}\) long and \(2.0 \mathrm{~m}\) wide. A concrete path of constant width is laid all the way around the shed. If the area of the path is \(9.50 \mathrm{~m}^{2}\) calculate its width to the nearest centimetre.

If the total surface area of a solid cone is \(486.2 \mathrm{~cm}^{2}\) and its slant height is \(15.3 \mathrm{~cm}\), determine its base diameter.

Determine the values of \(x\) and \(y\) which simultaneously satisfy the equations: \(y=5 x-4-2 x^{2}\) and \(y=6 x-7\)