Let's examine and complete the given quadratic formula for the general quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex]. We aim to fill in the missing part within the square root.
The quadratic formula is used to find the roots of the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex]. The complete quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Let's break down each component of the formula step-by-step:
1. Quadratic Equation Components:
- [tex]\( a \)[/tex] is the coefficient of [tex]\( x^2 \)[/tex].
- [tex]\( b \)[/tex] is the coefficient of [tex]\( x \)[/tex].
- [tex]\( c \)[/tex] is the constant term.
2. Quadratic Formula Terms:
- [tex]\( -b \)[/tex] is the negation of the linear coefficient.
- [tex]\( \sqrt{b^2 - 4ac} \)[/tex] is the square root of the discriminant of the quadratic equation.
- [tex]\( 2a \)[/tex] is twice the coefficient of [tex]\( x^2 \)[/tex].
3. Discriminant:
- The term inside the square root, [tex]\( b^2 - 4ac \)[/tex], is known as the discriminant.
- The discriminant determines the nature of the roots:
- If [tex]\( b^2 - 4ac > 0 \)[/tex], there are two distinct real roots.
- If [tex]\( b^2 - 4ac = 0 \)[/tex], there is one real root (repeated).
- If [tex]\( b^2 - 4ac < 0 \)[/tex], there are two complex roots.
Now, according to the question, we need to complete the quadratic formula by filling in the gap. The original incomplete part is:
[tex]\[ x = \frac{-b \pm \sqrt{\square}}{2a} \][/tex]
To fill in the gap correctly, we substitute the discriminant [tex]\( b^2 - 4ac \)[/tex] in place of the placeholder symbol (square). Hence, the complete formula is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Therefore, the correct expression to fill in the gap is:
[tex]\[ \boxed{b^2 - 4ac} \][/tex]
This completes the quadratic formula with the correct expression.
