If one of the roots of quadratic equation $7x^2 −50x + k$ = 0 is 7, then what is the value of k ?

Q: If one of the roots of quadratic equation $7x^2 −50x + k$ = 0 is 7, then what is the value of k ?

To determine the value of $ k $ in the quadratic equation $ 7x^2 - 50x + k = 0 $ given that one of its roots is 7, we can use the property of roots of quadratic equations. Specifically, if $ x = 7 $ is a root, then it satisfies the equation: \[ 7(7)^2 - 50(7) + k = 0 \] Let's calculate this step by step: 1. Calculate $ 7(7)^2 $: \[ 7 \times 49 = 343 \] 2. Calculate $ 50 \times 7 $: \[ 50 \times 7 = 350 \] 3. Substitute these values into the equation: \[ 343 - 350 + k = 0 \] 4. Simplify the equation: \[ -7 + k = 0 \] 5. Solve for $ k $: \[ k = 7 \] Thus, the value of $ k $ is 7, which corresponds to Option 1. Now, let's explain why the other options are incorrect: - **Option 2: 1** If $ k = 1 $, substituting back into the equation would not satisfy the condition that 7 is a root. Specifically, substituting $ k = 1 $ into the equation gives: \[ 343 - 350 + 1 = -6 \neq 0 \] Therefore, $ k = 1 $ is incorrect. - **Option 3: $\frac{50}{7}$** If $ k = \frac{50}{7} $, substituting back into the equation would not satisfy the condition that 7 is a root. Specifically, substituting $ k = \frac{50}{7} $ into the equation gives: \[ 343 - 350 + \frac{50}{7} \neq 0 \] This does not simplify to zero, hence $ k = \frac{50}{7} $ is incorrect. - **Option 4: $\frac{7}{50}$** If $ k = \frac{7}{50} $, substituting back into the equation would not satisfy the condition that 7 is a root. Specifically, substituting $ k = \frac{7}{50} $ into the equation gives: \[ 343 - 350 + \frac{7}{50} \neq 0 \] This does not simplify to zero, hence $ k = \frac{7}{50} $ is incorrect. Therefore, the only correct value for $ k $ that satisfies the condition of the problem is 7, which is Option 1.

Question

Accepted Answer

To determine the value of $ k $ in the quadratic equation $ 7x^2 - 50x + k = 0 $ given that one of its roots is 7, we can use the property of roots of quadratic equations. Specifically, if $ x = 7 $ is a root, then it satisfies the equation:

$$ 7(7)^2 - 50(7) + k = 0 $$

Let's calculate this step by step:

1. Calculate $ 7(7)^2 $:
   $$
   7 \times 49 = 343
   $$

2. Calculate $ 50 \times 7 $:
   $$
   50 \times 7 = 350
   $$

3. Substitute these values into the equation:
   $$
   343 - 350 + k = 0
   $$

4. Simplify the equation:
   $$
   -7 + k = 0
   $$

5. Solve for $ k $:
   $$
   k = 7
   $$

Thus, the value of $ k $ is 7, which corresponds to Option 1.

Now, let's explain why the other options are incorrect:

- **Option 2: 1**  
  If $ k = 1 $, substituting back into the equation would not satisfy the condition that 7 is a root. Specifically, substituting $ k = 1 $ into the equation gives:
  $$
  343 - 350 + 1 = -6 \neq 0
  $$
  Therefore, $ k = 1 $ is incorrect.

- **Option 3: $\frac{50}{7}$**  
  If $ k = \frac{50}{7} $, substituting back into the equation would not satisfy the condition that 7 is a root. Specifically, substituting $ k = \frac{50}{7} $ into the equation gives:
  $$
  343 - 350 + \frac{50}{7} \neq 0
  $$
  This does not simplify to zero, hence $ k = \frac{50}{7} $ is incorrect.

- **Option 4: $\frac{7}{50}$**  
  If $ k = \frac{7}{50} $, substituting back into the equation would not satisfy the condition that 7 is a root. Specifically, substituting $ k = \frac{7}{50} $ into the equation gives:
  $$
  343 - 350 + \frac{7}{50} \neq 0
  $$
  This does not simplify to zero, hence $ k = \frac{7}{50} $ is incorrect.

Therefore, the only correct value for $ k $ that satisfies the condition of the problem is 7, which is Option 1.