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: Try to factor the quadratic.
We need two numbers that multiply to give \(2 \times 6 = 12\) and add up to give \(-7\). Those numbers are \(-4\) and \(-3\), since \((-4)(-3) = 12\) and \(-4 + (-3) = -7\).
Rewrite the middle term using these numbers.
\[ 2x^2 - 4x - 3x + 6 = 0 \]

Step 2: Group and factor.
\[ 2x(x - 2) - 3(x - 2) = 0 \]
\[ (x - 2)(2x - 3) = 0 \]

Step 3: Solve each factor for x.
Either \(x - 2 = 0\), giving \(x = 2\), or \(2x - 3 = 0\), giving \(x = \frac{3}{2} = 1.5\).
So the two roots of the equation are 2 and 1.5.

Step 4: Pick the largest root.
Comparing 2 and 1.5, the larger value is 2. Option (c) 3 is not even a root of the equation, and option (d) is wrong since valid roots do exist.

Final Answer:
The largest value of x that satisfies the equation is 2. \[ \boxed{x = 2} \]