Question

Solve ${x^2} - 3x + 4$using quadratic formula.

Answer 1

Check if the given expression is arranged in the same form as the standard
expression,$a{x^2} + bx + c$. If not, then arrange it accordingly and compare the values. Recall the
quadratic formula. Put the compared values in the appropriate places to get the answer.

Complete step by step solution:
The quadratic formula is, $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$

Where ${b^2} - 4ac$= D, otherwise known as discriminant.

In the given question, we see that the expression is arranged in accordance with standard expression.

Comparing the expressions,
a = 1, b = -3 and c = 4

Now, putting the values in the quadratic formula we’ll get
$x = \;\dfrac{{3 \pm \sqrt {{3^2} - (4 \times 1 \times - 4)} }}{2}$
$\Rightarrow x = \;\dfrac{{3 \pm \sqrt {9 - ( - 16)} }}{2} \\\ \Rightarrow x = \;\dfrac{{3 \pm \sqrt {25} }}{2} \\\ \Rightarrow x = \dfrac{{3 \pm 5}}{2} \\\ \Rightarrow x = \dfrac{8}{2},\frac{{ - 2}}{2} \\\$

Thus, x will have 2 values, 4 and -1.

Note: There is another method of factorization that is called the “mid-term” factorization. It can be used only if the coefficient of x can be written as the sum of any two factors of the product of the constant and the coefficient of ${x^2}$ . It becomes easier to use this method only if one is dealing with integers. In case of fractional or imaginary values, this method is very cumbersome and hence should not be used. In such a case, you can always use the formula, $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$to find the factors.