}, \\; \\htmlData{anchor=coef-c}{\\color{@purple}{c = -24}}$"},{"latex":"$$x = \\frac{-\\htmlData{anchor=form-b1}{\\color{@amber}{b}} \\pm \\sqrt{\\htmlData{anchor=form-b2}{\\color{@amber}{b}}^2 - 4\\htmlData{anchor=form-a1}{\\color{@sky}{a}}\\htmlData{anchor=form-c1}{\\color{@purple}{c}}}}{2\\htmlData{anchor=form-a2}{\\color{@sky}{a}}}$"},{"latex":"$$x = \\frac{-(\\color{@amber}{5}) \\pm \\sqrt{(\\color{@amber}{5})^2 - 4(\\color{@sky}{1})(\\color{@purple}{-24})}}{2(\\color{@sky}{1})}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"coef-a"},{"color":"amber","style":"text-color","anchor":"coef-b"},{"color":"purple","style":"text-color","anchor":"coef-c"}],"connections":[{"to":"form-b1","from":"coef-b","color":"amber","label":"substitute b","style":"solid"},{"to":"form-a1","from":"coef-a","color":"sky","label":"substitute a","style":"solid"},{"to":"form-c1","from":"coef-c","color":"purple","label":"substitute c","style":"solid"}]},"order":3,"title":"Apply the quadratic formula","description":"We identify the coefficients of our quadratic equation and substitute them into the quadratic formula."},{"type":"text","order":4,"title":"Simplify and solve","content":"Now we simplify the expression inside the square root step by step:\n\n$$\\begin{aligned} x &= \\frac{-5 \\pm \\sqrt{25 - (-96)}}{2} \\\\ x &= \\frac{-5 \\pm \\sqrt{25 + 96}}{2} \\\\ x &= \\frac{-5 \\pm \\sqrt{121}}{2} \\end{aligned}$$\n\nSince $\\sqrt{121} = 11$, we obtain two possible values for $x$:\n\n$$\\begin{aligned} x_1 &= \\frac{-5 + 11}{2} = \\frac{6}{2} = 3 \\\\ x_2 &= \\frac{-5 - 11}{2} = \\frac{-16}{2} = -8 \\end{aligned}$$\n\n

This step earns the third method mark (M1).

"},{"type":"text","order":5,"title":"Find the valid length","content":"Since $x$ represents a physical side length of a triangle, it must be a positive number ($x > 0$).\n\nTherefore, we discard the negative solution $x = -8$.\n\nThe only valid value for $x$ is:\n$$x = 3$$\n\n

This completes the solution and awards the final accuracy mark (A1).

"}]}],"generatedAt":"2026-06-12T11:49:39.550Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":25},"walkthroughVideo":{"url":"https://assets.mathsgenie.co.uk/walkthroughs/videos/23957221-1cce-4d74-bc1a-87a0eafeb2fc.mp4","durationSeconds":131,"posterUrl":"https://assets.mathsgenie.co.uk/walkthroughs/videos/23957221-1cce-4d74-bc1a-87a0eafeb2fc.jpg"}},{"id":"a271722e-0045-4b48-a74b-a2c580a4da74","revisionId":"98e6ef60-c607-499c-91b5-d480310ce24e","questionNumber":null,"questionType":"LA","specification":"A triangle has two sides of length $7\\text{ cm}$ and $12\\text{ cm}$. The included angle between these sides is $x^\\circ$.","difficulty":2,"difficultyBand":"medium","totalMarks":4,"sourceTitle":null,"calculatorUse":"calc","isCalculation":true,"parts":[{"id":"15088401-7afb-4127-b4e1-601484edb82c","label":"a","content":"The area of the triangle is $30\\text{ cm}^2$.\n\nWork out the value of $x$.\n\nGive your answer to 3 significant figures.","markscheme":"- Sets up equation: $\\frac{1}{2} \\times 7 \\times 12 \\times \\sin(x) = 30$ M1\n- Simplifies to $42 \\sin(x) = 30$ M1\n- Rearranges to $\\sin(x) = \\frac{30}{42}$ M1\n- Calculates $x = 45.6$ A1","marks":4,"order":0,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Sets up the correct equation using the area formula for a triangle.","code":"M"},{"id":"p2","marks":1,"description":"Correctly simplifies the numerical product $0.5 \\times 7 \\times 12$.","code":"M"},{"id":"p3","marks":1,"description":"Rearranges the equation to isolate $\\sin(x)$.","code":"M"},{"id":"p4","marks":1,"description":"Calculates $x$ to 3 significant figures.","code":"A"}],"finalAnswers":["45.6"],"generatedAt":"2026-05-06T14:01:38.449Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"sine_area_formula","label":"Use the sine a

Finding the Area of Any Triangle

Question 3

Finding the Area of Any Triangle Questions

Finding the Area of Any Triangle

Question 3

Finding the Area of Any Triangle Questions