Find the roots of the quadratic equations by using the quadratic formula in each of the following:
(i) \(2x^2 - 3x - 5 = 0\)
(ii) \(5x^2 + 13x + 8 = 0\)
(iii) \(-3x^2 + 5x + 12 = 0\)
(iv) \(-x^2 + 7x - 10 = 0\)
(v) \(x^2 + 2\sqrt{2}\,x - 6 = 0\)
(vi) \(x^2 - 3\sqrt{5}\,x + 10 = 0\)
(vii) \(\dfrac{1}{2}x^2 - \sqrt{11}\,x + 1 = 0\)
(i) \(x=\dfrac{5}{2}\), \(x=-1\)
(ii) \(x=-1\), \(x=-\dfrac{8}{5}\)
(iii) \(x=3\), \(x=-\dfrac{4}{3}\)
(iv) \(x=5\), \(x=2\)
(v) \(x=\sqrt{2}\), \(x=-3\sqrt{2}\)
(vi) \(x=2\sqrt{5}\), \(x=\sqrt{5}\)
(vii) \(x=\sqrt{11}+3\), \(x=\sqrt{11}-3\)
We will solve all problems using the quadratic formula:
\[ x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a\), \(b\), and \(c\) are the coefficients of \(x^2\), \(x\), and the constant term in the equation.
\(D = b^2 - 4ac\) is called the discriminant.
Identify: \(a = 2\), \(b = -3\), \(c = -5\).
Discriminant: \(D = (-3)^2 - 4(2)(-5) = 9 + 40 = 49\).
Formula: \(x = \dfrac{-(-3) \pm \sqrt{49}}{2(2)} = \dfrac{3 \pm 7}{4}\).
So, \(x = \dfrac{10}{4} = \dfrac{5}{2}\) or \(x = \dfrac{-4}{4} = -1\).
Identify: \(a = 5\), \(b = 13\), \(c = 8\).
Discriminant: \(D = 13^2 - 4(5)(8) = 169 - 160 = 9\).
Formula: \(x = \dfrac{-13 \pm \sqrt{9}}{2(5)} = \dfrac{-13 \pm 3}{10}\).
So, \(x = -1\) or \(x = -\dfrac{8}{5}\).
Multiply through by -1 to make \(a\) positive: \(3x^2 - 5x - 12 = 0\).
Now, \(a = 3\), \(b = -5\), \(c = -12\).
Discriminant: \(D = (-5)^2 - 4(3)(-12) = 25 + 144 = 169\).
Formula: \(x = \dfrac{5 \pm \sqrt{169}}{6} = \dfrac{5 \pm 13}{6}\).
So, \(x = 3\) or \(x = -\dfrac{4}{3}\).
Multiply through by -1: \(x^2 - 7x + 10 = 0\).
Now, \(a = 1\), \(b = -7\), \(c = 10\).
Discriminant: \(D = (-7)^2 - 4(1)(10) = 49 - 40 = 9\).
Formula: \(x = \dfrac{7 \pm \sqrt{9}}{2} = \dfrac{7 \pm 3}{2}\).
So, \(x = 5\) or \(x = 2\).
Here, \(a = 1\), \(b = 2\sqrt{2}\), \(c = -6\).
Discriminant: \(D = (2\sqrt{2})^2 - 4(1)(-6) = 8 + 24 = 32\).
\(\sqrt{32} = 4\sqrt{2}\).
Formula: \(x = \dfrac{-2\sqrt{2} \pm 4\sqrt{2}}{2}\).
So, \(x = \sqrt{2}\) or \(x = -3\sqrt{2}\).
Here, \(a = 1\), \(b = -3\sqrt{5}\), \(c = 10\).
Discriminant: \(D = (-3\sqrt{5})^2 - 4(1)(10) = 45 - 40 = 5\).
Formula: \(x = \dfrac{3\sqrt{5} \pm \sqrt{5}}{2}\).
Factor out \(\sqrt{5}\): \(x = \dfrac{\sqrt{5}(3 \pm 1)}{2}\).
So, \(x = 2\sqrt{5}\) or \(x = \sqrt{5}\).
Multiply through by 2: \(x^2 - 2\sqrt{11}x + 2 = 0\).
Now, \(a = 1\), \(b = -2\sqrt{11}\), \(c = 2\).
Discriminant: \(D = (-2\sqrt{11})^2 - 4(1)(2) = 44 - 8 = 36\).
\(\sqrt{36} = 6\).
Formula: \(x = \dfrac{2\sqrt{11} \pm 6}{2} = \sqrt{11} \pm 3\).
So, \(x = \sqrt{11}+3\) or \(x = \sqrt{11}-3\).