--- title: "Simultaneous Equations with One Quadratic" description: "Solve a linear and a quadratic equation together by substituting the linear one into the quadratic, and read the answers as the points where a line meets a curve." canonical: https://lightmysky.com/learn/mathematics/simultaneous-equations-with-one-quadratic-mt_D89ql2sJvc source: https://lightmysky.com/learn/mathematics/simultaneous-equations-with-one-quadratic-mt_D89ql2sJvc.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Simultaneous Equations with One Quadratic Solve a linear and a quadratic equation together by substituting the linear one into the quadratic, and read the answers as the points where a line meets a curve. Subject: Mathematics · Area: Algebra · Ages 15 to 16 Page: https://lightmysky.com/learn/mathematics/simultaneous-equations-with-one-quadratic-mt_D89ql2sJvc ## Ready when they can - Substitute the linear equation into the quadratic and collect terms to one side - Give the answers as coordinate pairs, not just x values - Say what a single repeated solution means about the line and the curve ## Lesson: Simultaneous equations with one quadratic You have sketched parabolas. A nonvertical line and a quadratic graph meet where both equations hold. Each solution is a coordinate pair (x, y), not an x-value alone. Substitution first finds intersection x-coordinates; substitute each into either equation to find y. If y = ax² + bx + c and y = mx + n, equate their right sides and subtract mx + n: ax² + (b - m)x + c - n = 0. Check both subtractions before solving. For y = x² - 14 and y = 2x + 1, equate the right sides: x² - 14 = 2x + 1. Collect to get x² - 2x - 15 = 0. Find two numbers with product -15 and sum -2: -5 and 3. Factorise as (x - 5)(x + 3) = 0. Set either factor to zero, giving x = 5 or x = -3. These are the intersection x-coordinates. **Example.** The discriminant of ax² + bx + c = 0 is D = b² - 4ac. For x² + 6x + 9 = 0, a = 1, b = 6 and c = 9. Thus D = 36 - 36 = 0. A negative discriminant gives no real roots. For x² + 4x + 7 = 0, D = 16 - 28 = -12. No real x makes the two substituted expressions equal, so there are no real intersections. For a known intersection x-coordinate, use the line to find y. If y = 3x + 2 and x = -4, then y = -12 + 2 = -10. Keep the values together as (-4, -10). Both equations give the same y at a true intersection. Use whichever has less arithmetic for the particular value. For a line already written as y = mx + n, direct substitution is often convenient, but it is not a universal rule. After substitution, roots describe all possible intersection x-coordinates. If D < 0, there are none. This conclusion concerns real intersections and does not require attempting factorisation. **Example.** When the collected equation is monic, look for two numbers whose product is its constant and whose sum is its x coefficient. x² - 7x + 10 = (x - 2)(x - 5), giving roots 2 and 5. **Example.** Solve y = x² - 11 and y = 4x + 1. Equating gives x² - 4x - 12 = 0. Numbers -6 and 2 have product -12 and sum -4, so factorise as (x - 6)(x + 2) = 0. Thus x = 6 or -2. The line gives y = 25 or -7. The intersection points are (6, 25) and (-2, -7). Count distinct real roots of the substituted quadratic to count intersections: D > 0 gives two, D = 0 gives one and D < 0 gives none. A repeated root is one x-coordinate, not two distinct points. **Example.** For y = x² and y = 6x - 9, substitution gives (x - 3)² = 0. There is one root, x = 3, and y = 9. The nonvertical line is tangent to the quadratic at (3, 9). **Example.** To reconstruct a line through two points, calculate m = (y₂ - y₁)/(x₂ - x₁), then find c in y = mx + c. Through (0, 2) and (2, 8), m = 6/2 = 3 and c = 2, so y = 3x + 2. To find a sum of intersection x-coordinates, first obtain both roots. For y = x² - 19 and y = 2x + 5, equating and collecting gives x² - 2x - 24 = 0. Numbers -6 and 4 have product -24 and sum -2, so (x - 6)(x + 4) = 0. The roots are 6 and -4. Add them with their signs: 6 + (-4) = 2. Do not add y-coordinates or absolute values. When both equations contain an x term, collect that term carefully. Equating x² + bx + c with mx + n changes the x coefficient to b - m and the constant to c - n. Then factorise or use the formula. For a nonvertical tangent to a quadratic-function graph, substitution gives a quadratic with a repeated root. A quadratic has no further distinct root, so this line meets that parabola only at the tangent point. This claim does not apply to every kind of curve. **Example.** For ax² + bx + c = 0, use x = (-b ± √(b² - 4ac))/(2a). For x² - 4x - 2 = 0, x = (4 ± √24)/2 = 2 ± √6. Evaluate the requested sign before rounding. You can establish that a line misses a quadratic by substitution. Keep the signs when calculating D. For x² - 2x + 5 = 0, D = 4 - 20 = -16, proving that there are no real intersections. Check a proposed intersection by substituting both coordinates into both original equations. Even with a correct x-value, the y-value must also be correct. If a point fails either equation, replace or reject that proposed pair. A downward quadratic can be solved with the same method. After equating it with the line, multiplying the entire equation by -1 may make factorisation easier. This changes neither its roots nor the corresponding intersection points. **Example.** For y = x² and y = 3x + k, substitution gives x² - 3x - k = 0. Tangency requires D = 9 + 4k = 0, so k = -9/4. Find the parameter before finding the tangent point. A line with gradient m and y-intercept c has equation y = mx + c. Build that equation from the given information before substituting into the quadratic. A point (0, c) gives the intercept directly. **Example.** For y = x + c and y = x² - 5x + 10, substitution gives x² - 6x + 10 - c = 0. No real intersections require D = 36 - 4(10 - c) < 0. Hence c < 1; the largest allowed integer is 0. **Example.** Find the tangent to y = x² - 10x + 21 at (3, 0). Write y = m(x - 3). Substitution gives x² - (10 + m)x + 21 + 3m = 0. A repeated root at 3 means this equals (x - 3)². Compare coefficients: 10 + m = 6, so m = -4; also 21 + 3m = 9. The line is y = -4x + 12. You can reconstruct a system from its collected equation. If the equation is Q(x) - L(x) = 0, it comes from y = Q(x) and y = L(x). Expand and collect to check a proposed pair of original rules. To test supplied candidate points, substitute each into both equations rather than solving again. Keep only pairs for which both equalities hold. A point on the line alone is not enough. Substitution can eliminate x instead of y. If x = y - 2 and x = y² - 8, equating gives y² - y - 6 = 0. Solve for y, then substitute into x = y - 2. The final pair still uses the order (x, y). For a horizontal line y = k and a quadratic y = (x - h)² + v, substitution gives (x - h)² = k - v. A positive right side gives two intersections, zero gives tangency, and a negative right side gives none. **Recap.** Substitute, collect terms and solve; then find y for each x. Two distinct real roots give two points, a repeated root gives tangency, and a negative discriminant gives no real meeting points. *Lesson and assessment explanations written for Marble.* ## Practice 29 questions on this page, each with its working shown. ## Needs first - [Simultaneous equations](https://lightmysky.com/learn/mathematics/simultaneous-equations-mt_mqgu72aCMz) - [Sketching Quadratic Graphs from Roots and the Turning Point](https://lightmysky.com/learn/mathematics/sketching-quadratic-graphs-from-roots-and-the-turning-point-mt_RcrJIbbN2x) - [Solving quadratic equations by factorising](https://lightmysky.com/learn/mathematics/solving-quadratic-equations-by-factorising-mt_tg_sqRzDjU) ## Opens up - [Areas Between Two Curves](https://lightmysky.com/learn/mathematics/areas-between-two-curves-mt_lDBZmnLK1b)