To solve the given quadratic equations and convert them to standard form $ax^2 + bx + c = 0$, we can follow these steps:

1. Given Equation: $4x^2 - 11x = 7$

   • Subtract 7 from both sides to set the equation to zero: $4x^2 - 11x - 7 = 0$
   • The standard form is $4x^2 - 11x - 7 = 0$, with $a = 4$, $b = -11$, and $c = -7$.

2. Given Equation: $5x^2 = 3x$

   • Subtract $3x$ from both sides to set the equation to zero: $5x^2 - 3x = 0$
   • The standard form is $5x^2 - 3x = 0$, with $a = 5$, $b = -3$, and $c = 0$.

3. Given Equation: $(x + 3)(x - 2) = 1$

   • Expand the left-hand side: $x^2 - 2x + 3x - 6 = x^2 + x - 6$
   • Subtract 1 from both sides to set the equation to zero: $x^2 + x - 7 = 0$
   • The standard form is $x^2 + x - 7 = 0$, with $a = 1$, $b = 1$, and $c = -7$.

4. Given Equation: $(x - 7)^2 + 3 = 0$

   • Expand $(x - 7)^2$: $(x - 7)^2 = x^2 - 14x + 49$
   • Substitute back into the equation: $x^2 - 14x + 49 + 3 = x^2 - 14x + 52 = 0$
   • The standard form is $x^2 - 14x + 52 = 0$, with $a = 1$, $b = -14$, and $c = 52$.

5. Given Equation: $6x^2 = 5x - 4$

   • Subtract $5x - 4$ from both sides to set the equation to zero: $6x^2 - 5x + 4 = 0$
   • The standard form is $6x^2 - 5x + 4 = 0$, with $a = 6$, $b = -5$, and $c = 4$.

Here is the completed table:

GIVEN	STANDARD FORM	a	b	c
$4x^2 - 11x = 7$	$4x^2 - 11x - 7 = 0$	4	-11	-7
$5x^2 = 3x$	$5x^2 - 3x = 0$	5	-3	0
$(x + 3)(x - 2) = 1$	$x^2 + x - 7 = 0$	1	1	-7
$(x - 7)^2 + 3 = 0$	$x^2 - 14x + 52 = 0$	1	-14	52
$6x^2 = 5x - 4$	$6x^2 - 5x + 4 = 0$	6	-5	4

Would you like any further details or have any questions?

Here are 8 related questions:

1. How do you factor quadratic equations?
2. What methods can be used to solve quadratic equations?
3. How does the discriminant of a quadratic equation determine the number of solutions?
4. Can you give an example of solving a quadratic equation by completing the square?
5. How do you graph a quadratic equation?
6. What is the significance of the vertex in a quadratic equation?
7. How do you determine the axis of symmetry for a quadratic function?
8. How can quadratic equations be applied in real-life scenarios?

Tip: Always ensure to set the quadratic equation to zero before identifying the coefficients $a$, $b$, and $c$.