Question

$\rm Derive \ the \ quadratic \ formula$

$\rm \dfrac{-b \pm \sqrt{b^2 -4ac}}{2a}$

Answer in copy or use latex.​

Answer 1

We'll use the method of "Completing the Square" to derive the quadratic equation.

What does Completing the Square mean?

When we say we're completing the square of an equation we're basically expressing it's LHS as a perfect square trinomial.

Now, we know that a quadratic equation is of the form:

$\rm \Longrightarrow ax^2 + bx + c = 0$

Step 1: Make x² independent of a.

For this, let's divide both sides by a.

$\rm \Longrightarrow \dfrac{ax^2}{a} \ + \ \dfrac{bx}{a} \ + \ \dfrac{c}{a} \ = \ \dfrac{0}{a}$

$\rm \Longrightarrow x^2 \ + \ \dfrac{bx}{a} \ + \ \dfrac{c}{a} \ = 0$

Step 2: Transpose c/a to the RHS.

$\rm \Longrightarrow x^2 \ + \ \dfrac{bx}{a} \ = \ - \dfrac{c}{a}$

Step 3: Focus on the coefficient of x, which is b/a. Now, divide b/a by 2, we get b/2a. Now square b/2a, we get (b/2a)², add this to both sides of the equation to it remains balanced.

$\rm \Longrightarrow x^2 \ + \ \dfrac{bx}{a} \ + \Bigg(\dfrac{b}{2a}\Bigg)^2 = \ - \dfrac{c}{a} + \Bigg(\dfrac{b}{2a}\Bigg)^2$

Step 4: We can see that the LHS is expressible in the form of (p + q)² = p² + q² + 2pq. Where p = x and q = b/2a.

$\rm \Longrightarrow \Bigg(x + \dfrac{b}{2a}\Bigg)^2 = \ - \dfrac{c}{a} + \Bigg(\dfrac{b}{2a}\Bigg)^2$

$\rm \Longrightarrow \Bigg(x + \dfrac{b}{2a}\Bigg)^2 = \ - \dfrac{c}{a} + \dfrac{b^2}{4a^2}$

Step 5: Taking LCM of 4a² & a we get.

$\rm \Longrightarrow \Bigg(x + \dfrac{b}{2a}\Bigg)^2 = \ \dfrac{b^2}{4a^2} - \dfrac{c}{a}$

$\rm \Longrightarrow \Bigg(x + \dfrac{b}{2a}\Bigg)^2 = \ \dfrac{b^2 - 4ac}{4a^2}$

$\rm \Longrightarrow x + \dfrac{b}{2a} = \ \pm \ \sqrt{\dfrac{b^2 - 4ac}{4a^2}}$

$\rm \Longrightarrow x + \dfrac{b}{2a} = \ \dfrac{\sqrt{ \ b^2 - 4ac \ }}{2a}$

$\rm \Longrightarrow x = - \dfrac{b}{2a} \pm \dfrac{\sqrt{ \ b^2 - 4ac \ }}{2a}$

$\rm \Longrightarrow x = \dfrac{-b \pm \sqrt{ \ b^2 - 4ac \ }}{2a}$

Hence proved.

Answer 2

We'll use the method of "Completing the Square" to derive the quadratic equation.

What does Completing the Square mean?

When we say we're completing the square of an equation we're basically expressing it's LHS as a perfect square trinomial.

Now, we know that a quadratic equation is of the form:

$⟹ax2+bx+c=0$

step 1 : Make x² independent of a.

For this, let's divide both sides by a

$⟹aax2 + abx + ac = a0</p><p>$

$⟹x2 + abx + ac =0$

step 2: Transpose c/a to the RHS.

$⟹x2 + abx = −ac</p><p>$

Step 3: Focus on the coefficient of x, which is b/a. Now, divide b/a by 2, we get b/2a. Now square b/2a, we get (b/2a)², add this to both sides of the equation to it remains balanced.

$⟹x=2a−b± b2−4ac </p><p>$