• Quick retrieval from earlier learning (The quadratic formula; Simplify surds (radicals)). Students answer on mini whiteboards.
• Show answers together and note who needs support in the main task.

1. Solve 4x² − 9x − 8 = 0 using the quadratic formula. Give the smaller solution to 2 decimal places. → -0.68Prompt: x = (−b ± √(b² − 4ac)) / 2a.
2. Simplify √80. It can be written as k√5. What is k? → 4Prompt: Look for the largest square number that divides it.
3. How many real solutions does 5x² − x − 9 = 0 have? (Two / One (repeated) / None) → TwoPrompt: Work out the discriminant b² − 4ac.

• Show a "think of a number" puzzle and ask the class to work out the starting number.
• Share the objective: Give exact solutions with the quadratic formula. Read the success criteria together.

• Work through the example out loud, saying why you do each step, not just what you do.
• Example: Solve 2x² − 3 = 0. Give exact answers in the form (p ± √q)/r.
• Step 1: b² − 4ac = 24
• Step 2: x = (0 ± √24)/4
• Answer: x = (0 ± √24)/4
• Pause after each step and ask a student to say what comes next.

• Do these together. Students try each step on whiteboards before you reveal it.
• Use the hints as prompts rather than giving the answer.

1. Solve x² − x − 4 = 0. Give exact answers in the form (p ± √q)/r. (x = 1 ± √17/2 / x = (−1 ± √17)/2 / x = (1 ± √17)/2 / x = (1 ± √17)/1) → x = (1 ± √17)/2Prompt: x = (−b ± √(b² − 4ac))/(2a).
2. Solve x² − 10x + 5 = 0 with the quadratic formula. Simplify fully. (x = 10 ± 2√5 / x = 5 ± √20/2 / x = 5 ± 2√5 / x = 5 ± 4√5) → x = 5 ± 2√5Prompt: The discriminant is 80: simplify √80.

• Students work on the student sheet. Circulate and check the first two answers of every student early.
• Students who finish: write a question of their own for a partner, with the answer.

The 8 questions are on the student sheet below, with answers in the key.

• Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.

1. Solve x² − 10x + 7 = 0 with the quadratic formula. Simplify fully. (x = 10 ± 3√2 / x = 5 ± 3√2 / x = 5 ± 6√2 / x = 5 ± √18/2) → x = 5 ± 3√2Prompt: The discriminant is 72: simplify √72.
2. Solve 3x² + 8x + 2 = 0. Give exact answers in the form (p ± √q)/r. (x = (−8 ± √40)/6 / x = (−8 ± √88)/6 / x = (8 ± √40)/6 / x = (−8 ± √40)/3) → x = (−8 ± √40)/6Prompt: x = (−b ± √(b² − 4ac))/(2a).

Free online practice with hints and printable worksheets for this skill: eurekami.com/math/grade-10.