Problem 105 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. The quadratic equation \(-3 x^{2}-x=10\) has two real solutions.

Step-by-Step Solution

Answer

The statement is false. The quadratic equation \( -3x^{2} - x -10 = 0\) does not have two real solutions, but rather has two complex roots because the discriminant is negative.

Step 1 - Write the equation in standard form

The quadratic equation \(-3 x^{2}-x=10\) should be written in standard form \(ax^{2} + bx + c = 0\). This can be done by moving all terms to the left-hand side to get \( -3x^{2} - x -10 = 0 \)

Step 2 - Calculate the discriminant

The discriminant (D) of the quadratic equation is a key in the determination of the nature of roots of the equation. It is given by the formula \(D = b^{2} - 4ac\). In this example it gives \( D = (-1)^{2} - 4*(-3)*(-10) = 1 - 120 = -119\)

Step 3 - Evaluate the nature of roots

If the discriminant is positive, the quadratic equation has two distinct real roots. If it is zero, it has exactly one real root (a repeated root). If it is negative, it has two complex roots. In this case, since \(D = -119\) which is less than zero, the equation has two complex roots.