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Solving Simple Quadratic Equations

Learn how to solve simple quadratic equations, like x2=kx^2 = k and x2−dx=0x^2−d x = 0, and find all possible solutions. Let’s get started! 🚀

Solving Simple Quadratic Equations - introduction visual

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Solving the quadratic equation x² = k, showing the general solution x equals plus or minus the square root of k, and example of x² = 9 with solutions.Solving quadratic equations with the steps to isolate x² and find solutions x sub one equals the positive square root of k.Solving quadratic equations, demonstrating x² = k and x² + 50 = 0 leading to no real solution since x² = -25.Solving quadratic equations by factorisation, showing x² - dx = 0 with solutions x sub one equals zero, and x sub two equals d.Solving quadratic equation x² - dx = 0 with steps to simplify and factorise, showing solutions x₁ = 0 and x₂ = 5.Solving simple quadratic equations: x² = k gives x = ±√k, and x² − dx = 0 factors to x = 0 or x = d.

🛎️ Solving Equations of the Form x2=kx^2 = k

  • If x2=kx^2 = k, take the square root to get x=±kx = \pm\sqrt{k}. Important: this only works when k≥0k \ge 0.
  • For example, solving x2=9x^2 = 9 gives x=±9x = \pm\sqrt{9}, so the solutions are x=3x = 3 and x=−3x = -3.

🛎️ Example: Solving 3x2−12=03x^2 - 12 = 0

  • Rearrange by adding 12 to both sides so it becomes 3x2=123x^2 = 12, which is now in the form x2=kx^2 = k.
  • Divide both sides by 3 to simplify and get x2=4x^2 = 4.
  • Take the square root to get x=±2x = \pm 2, so the solutions are x=2x = 2 and x=−2x = -2.

🛎️ When There Is No Real Solution

  • If x2x^2 equals a negative number, there is no real solution because no real number squares to a negative.
  • For example, x2=−25x^2 = -25 has no real solution, so −5-5 is not an answer.

🛎️ Solving Equations of the Form x2−dx=0x^2 - dx = 0

  • Factorise to get x(x−d)=0x(x - d) = 0.
  • Set each factor equal to zero to get x=0x = 0 and x=dx = d.

🛎️ Example: 5x2=25x5x^2 = 25x

  • First make the coefficient of x2x^2 equal to 1 by dividing by 5, which helps to simplify the calculations.
  • Factorise to get x(x−5)=0x(x - 5) = 0, so the solutions are x=0x = 0 or x=5x = 5.

🛎️ Solving Simple Quadratic Equations (Summary)

  • If an equation is in the form x2=kx^2 = k, the solutions are x=±kx = \pm\sqrt{k} and this only works when k≥0k \ge 0.
  • If an equation is in the form x2−dx=0x^2 - dx = 0, factorise to x(x−d)=0x(x-d)=0 and the solutions are x=0x=0 and x=dx=d.

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