Question

Solve \(\frac{1}{2} x^{2}+5 x=28\) by multiplying first by 2 and then using al- Khw?rizmi's procedure. Similarly, solve \(2 x^{2}+\) \(10 x=48\) by first dividing by 2

Short answer

Equation 1: \(\frac{1}{2}x^2 + 5x = 28\) Equation 2: \(2x^2 + 10x = 48\) Answer: For Equation 1, the solutions are \(x = -14\) and \(x = 4\). For Equation 2, the solutions are \(x = -8\) and \(x = 3\).

Step-by-step solution

Equation 1: Multiply both sides by 2

To get rid of the fraction, multiply both sides of the equation by 2: \(2(\frac{1}{2}x^2 + 5x) = 2(28)\) This simplifies to the quadratic equation: \(x^2 + 10x = 56\)

Equation 1: Apply al-Khwārizmī's procedure

Step 1: Move the constant term (56) to the right side of the equation: \(x^2 + 10x - 56 = 0\) Step 2: Find two numbers whose product is -56 and sum is 10. These numbers are 14 and -4. Step 3: Rewrite the equation like this: \((x + 14)(x - 4) = 0\) Step 4: Find the values of x for which the equation is true: \(x = -14,\; x = 4\) So, the solutions for Equation 1 are \(x = -14\) and \(x = 4\).

Equation 2: Divide both sides by 2

To simplify the equation, divide both sides by 2: \(\frac{2x^2 + 10x}{2} = \frac{48}{2}\) This simplifies to the quadratic equation: \(x^2 + 5x = 24\)

Equation 2: Apply al-Khwārizmī's procedure

Step 1: Move the constant term (24) to the right side of the equation: \(x^2 + 5x - 24 = 0\) Step 2: Find two numbers whose product is -24 and sum is 5. These numbers are 8 and -3. Step 3: Rewrite the equation like this: \((x + 8)(x - 3) = 0\) Step 4: Find the values of x for which the equation is true: \(x = -8,\; x = 3\) So, the solutions for Equation 2 are \(x = -8\) and \(x = 3\).

Key concepts

al-Khwārizmī's procedure

Al-Khwārizmī's procedure is a historical method for solving quadratic equations, named after the Persian mathematician Muhammad ibn Musa al-Khwārizmī. He is credited with bringing the concept of algorithm to mathematics. His method is a systematic procedure for solving equations that involve unknown values. In this technique, we work to rearrange an equation to the form of \(ax^2 + bx + c = 0\).

Here's how al-Khwārizmī’s process is typically applied:

• Start by moving all terms to one side of the equation so that the equation is equal to zero. This usually involves moving a constant term to the opposite side.
• Next, look for two numbers that multiply to equal the constant term \(c\) and add up to equal the middle coefficient \(b\).
• These numbers are then used to factor the quadratic expression into the format \((x + m)(x + n) = 0\).
• Lastly, set each factor equal to zero and solve for \(x\) to find the possible solutions.

Al-Khwārizmī's method may seem straightforward, but it remains a foundational concept used in modern algebra, aiding us in solving quadratic equations effortlessly.

Factoring quadratics

Factoring quadratics is a method used to solve quadratic equations that are set in the standard form \(ax^2 + bx + c = 0\). When factoring quadratics, the main goal is to express the quadratic equation as a product of two binomials.

Here's the step-by-step approach:

• Begin by identifying the coefficients \(a\), \(b\), and \(c\) in the quadratic equation.
• Find two numbers that such that they not only multiply to \(a \times c\) but also add up to \(b\).
• Once these numbers are found, write two binomials whose product gives the original quadratic equation.
• To confirm your factors are correct, you can expand or "foil" the binomials and check that they result in the original quadratic equation.

This method is incredibly handy for quickly finding the solutions of a quadratic equation, especially when you can easily spot the two numbers that satisfy the conditions for factoring effectively.

Solving quadratic equations

Solving quadratic equations involves finding the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). These equations can be solved using various methods, each suited for different forms of quadratic equations.

Here's a broader view of the main methods used to solve them:

• **Factoring**: If the quadratic can be easily factored, you can set each binomial factor equal to zero and solve for \(x\). This is often the simplest and quickest method when applicable.
• **Completing the square**: This involves manipulating the equation to form a perfect square trinomial, making it easier to solve by taking the square root of both sides.
• **Quadratic formula**: A universal method applicable to any quadratic equation: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This formula derives directly from completing the square and provides solutions even when factoring is not easy.

Each of these methods come with their own set of advantages and can be chosen based on the specific quadratic equation at hand, offering flexibility and multiple pathways to find the answers.