determine the nature of the roots for 16𝑥^2 + 25 = 40x
To determine the nature of the roots for the equation 16𝑥^2 + 25 = 40x, we can rearrange the equation into the standard form of a quadratic equation:
16𝑥^2 - 40𝑥 + 25 = 0
Now, we need to calculate the discriminant (𝐷) to determine the nature of the roots. The discriminant formula is:
𝐷 = 𝑏^2 - 4𝑎𝑐
In this case, 𝑎 = 16, 𝑏 = -40, and 𝑐 = 25. Substituting these values into the formula:
𝐷 = (-40)^2 - 4(16)(25)
𝐷 = 1600 - 1600
𝐷 = 0
If the discriminant is greater than 0 (𝐷 > 0), the quadratic equation has two distinct real roots.
If the discriminant is equal to 0 (𝐷 = 0), the quadratic equation has two identical real roots.
If the discriminant is less than 0 (𝐷 < 0), the quadratic equation has no real roots (only complex roots).
In this case, since the discriminant is equal to 0 (𝐷 = 0), the quadratic equation 16𝑥^2 + 25 = 40x has two identical real roots.