= \\frac{1}{2}x$$","highlights":[{"text":"two sides of length $x$ cm and $(x + 5)$ cm","location":"specification"},{"text":"The angle between these two sides is $30^{\\circ}$","location":"specification"}]},{"type":"text","order":1,"title":"Set up the area","content":"Now we can use the standard area formula for a triangle:\n$$\\text{Area} = \\frac{1}{2} \\times \\text{base} \\times \\text{height}$$\n\nSubstitute the base $b = x + 5$, height $h = \\frac{1}{2}x$, and the given area of $6\\text{ cm}^2$ into the formula:\n$$6 = \\frac{1}{2} (x + 5) \\left(\\frac{1}{2}x\\right)$$\n\nMultiply the terms on the right-hand side:\n$$6 = \\frac{1}{4} x(x + 5)$$\n\nTo eliminate the fraction, multiply both sides of the equation by $4$:\n$$24 = x(x + 5)$$

This earns the first two method marks (M1 M1) for setting up and simplifying the relationship.

","highlights":[{"text":"The area of the triangle is $6$ cm$^2$","location":"specification"}]},{"type":"text","order":2,"title":"Form a quadratic equation","content":"Next, we expand the brackets and rearrange the equation into standard quadratic form, $ax^2 + bx + c = 0$:\n\n$$\\begin{aligned} 24 &= x^2 + 5x \\\\ 0 &= x^2 + 5x - 24 \\end{aligned}$$\n\nThus, our quadratic equation is:\n$$x^2 + 5x - 24 = 0$$"},{"type":"hyper_latex","block":{"nodes":[{"text":"We solve the quadratic equation $x^2 + 5x - 24 = 0$ using the quadratic formula with:"},{"latex":"$$\\htmlData{anchor=coef-a}{\\color{@sky}{a = 1}}, \\; \\htmlData{anchor=coef-b}{\\color{@amber}{b = 5}}, \\; \\htmlData{anchor=coef-c}{\\color{@purple}{c = -24}}$"},{"latex":"$$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-87a0eafeb2f

Finding the Area of Any Triangle

Question 2

Finding the Area of Any Triangle Questions

Finding the Area of Any Triangle

Question 2

Finding the Area of Any Triangle Questions