Quadratic Equations & Non-Linear Equations

Solve x² + x − 12 = 0 by factorisation.

Solve x² − 4x − 3 = 0 by completing the square, giving your answer in surd form.

Solve 2x² + 3x − 4 = 0 using the quadratic formula, giving your answer in surd form.

Solve simultaneously: y = x + 1 and y = x² − 5.

Solve simultaneously: y = x − 1 and x² + y² = 25.

Solve simultaneously: y = x² and y = 3x − 2.

Solve simultaneously: y = x² and y = −3x + 10.

Solve simultaneously: y = x² and y = 5x − 4.

Solve simultaneously: y = x² + 2x and y = −3x − 6.

Solve simultaneously: y = x + 2 and x² + y² = 20.

Solve simultaneously: y = x − 1 and x² + y² = 13.

Solve simultaneously: y = x² − 4x and y = x + 6.

Solve simultaneously: y = 4 and y = x² + x − 2.

Solve simultaneously: y = 2x − 5 and x² + y² = 25.

Solve simultaneously: y = x² − 6x + 5 and y = −2x + 5.

Solve simultaneously: y = 3x − 10 and x² + y² = 10. (Careful — check how many solutions this one really has.)

A rectangle has a length 4 cm more than its width. Its area is 96 cm².

Let the width be x cm. Form a quadratic equation in x, and solve it to find the dimensions of the rectangle.

The product of two consecutive positive integers is 156. Find the two integers.

A cyclist travels 60 km at a steady speed. If she had travelled 5 km/h faster, the journey would have taken 1 hour less.

Let her original speed be x km/h. Form an equation in x, and solve it to find her original speed.

A right-angled triangle has a shorter leg that is 7 cm shorter than the other leg, and a hypotenuse of 13 cm.

Let the shorter leg be x cm. Form an equation in x, and solve it to find the lengths of both legs.

A rectangular garden is 5 m longer than it is wide. Its area is 84 m².

Let the width be x m. Form a quadratic equation and solve it to find the dimensions.

A number added to its own square gives 42. Find the possible value(s) of the number.

Two positive numbers differ by 3. Their product is 88. Find the two numbers.

The product of two consecutive even numbers is 168. Find the two numbers.

A picture frame is 3 cm wider than it is tall, and has an area of 154 cm².

Let the height be h cm. Form a quadratic equation and solve it to find the height and width.

Priya is currently x years old. In 3 years, the product of her age then and her current age will be 180.

Form a quadratic equation and solve it to find Priya’s current age.

A boat travels 24 km upstream and 24 km back downstream on a river flowing at 2 km/h. The whole trip takes 5 hours.

Let the boat’s speed in still water be x km/h. Form an equation in x, and solve it to find x.

Two pipes together fill a tank in 12 hours. Working alone, one pipe would take 10 hours longer than the other.

Let the faster pipe’s time (working alone) be x hours. Form an equation in x, and solve it to find how long each pipe takes alone.

A shop buys x identical calculators for a total of $240. If each calculator had cost $2 less, the shop could have bought 4 more calculators for the same total amount.

Form an equation in x, and solve it to find x.

A 5 m ladder leans against a vertical wall. The foot of the ladder is 1 m further from the wall than the height it reaches up the wall.

Let the height reached be x m. Form an equation in x, and solve it to find the height and the distance of the foot from the wall.