Question

If \(2x^2 - 7x + 6 = 0\), what is the largest value of \(x\) which satisfies the equation?

• A. 2
• B. \(\frac{3}{2}\)
• C. 3
• D. None of these

Hint:

Factor the quadratic into two linear factors, find both roots, and pick the larger one.

Answer 1

The Correct Option is A

Step 1: Identify the coefficients for the quadratic formula.
For $2x^2 - 7x + 6 = 0$, matching with the standard form $ax^2+bx+c=0$, we get $a=2$, $b=-7$, $c=6$.

Step 2: Write the quadratic formula.
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Step 3: Substitute the values and simplify the discriminant.
$b^2 - 4ac = (-7)^2 - 4(2)(6) = 49 - 48 = 1$.
Since the discriminant is 1, a positive perfect square, the equation has two distinct real roots.
\[ x = \frac{7 \pm \sqrt{1}}{4} = \frac{7 \pm 1}{4} \]

Step 4: Work out both roots.
Taking the plus sign: $x = \frac{7+1}{4} = \frac{8}{4} = 2$.
Taking the minus sign: $x = \frac{7-1}{4} = \frac{6}{4} = 1.5$.

Step 5: Choose the larger root.
Between 2 and 1.5, 2 is larger, so this is the answer, matching what factoring gave as well.

Final Answer:
Using the quadratic formula also confirms the largest root is 2. \[ \boxed{x = 2} \]