Understanding Equal Roots in Quadratic Equations

A quadratic equation is an equation of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are coefficients and $a \neq 0$.

For a quadratic equation to have equal roots (also called a repeated root or a single real root), its discriminant must be equal to zero.

Calculating the Discriminant

The discriminant, denoted by $D$, is calculated using the formula:

$$ D = b^2 - 4ac $$

Applying the Condition to the Given Equation

The given quadratic equation is:

$$ 7x^2 - 4x + k = 0 $$

By comparing this equation with the standard form $ax^2 + bx + c = 0$, we can identify the coefficients:

• $a = 7$
• $b = -4$
• $c = k$

Solving for k for Equal Roots

For the equation to have equal roots, the discriminant must be zero ($D=0$).

Substitute the values of $a$, $b$, and $c$ into the discriminant formula:

$$ D = (-4)^2 - 4(7)(k) $$

Simplify the expression:

$$ D = 16 - 28k $$

Now, set the discriminant to zero:

$$ 16 - 28k = 0 $$

To find the value of k, rearrange the equation:

$$ 16 = 28k $$

Isolate $k$ by dividing both sides by 28:

$$ k = \frac{16}{28} $$

Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 4:

$$ k = \frac{16 \div 4}{28 \div 4} $$

$$ k = \frac{4}{7} $$

Conclusion

Therefore, the value of k that makes the quadratic equation $7x^2 - 4x + k = 0$ have equal roots is $\frac{4}{7}$.